Several major factors play a big role in level 1 fights.
The base statistics of the champion
Since at level one, the tanks hasn't had a chance to build any armor, magic resist, or health yet, or the carries hasn't had a chance to build any damage items yet, everything hinges on what base stats champions are given.
When looking at the base stats, it is also important to pay attention specifically to the starting stat plus the stat gained per level. Let's look at Gangplank for example. Gangplank is generally considered a pretty squishy champion unless he builds health and armor/magic resist. He is one of those champions that people build as a glass cannon sometimes, someone who does a lot of damage but dies easily. However lets look at his base stat. He starts out with 495 + 81/level. That's a starting health of 576 plus any health that his starting item might have given him. That's quite a lot for someone who is considered a relatively squishy champion.
Now lets look at a champion that is normally considered tanky. Lets look at Udyr. Udyr's health is 427 + 99/level so he starts with 526. That is 50 less health! But Udyr has much higher health gained per level than Gangplank does. This means Gangplank has a stronger earlier game(in terms of health). You can take this thought process and apply it to the armor and magic resist and other statistics.
Skills and Passives optimized for level 1
This should be a philosophy that should apply to late game too but it is especially important early game because each of your teammates will only have on skill available to them instead of all 4. Choose a team that is balanced. You need a good balance of crowd control plus damage. A team that focuses too much on damage but has no crowd control will not be able to focus down on a single target enough to kill them. With the use of a snare or a stun, it provides a definite bull eyes sign for all your teammates to focus, usually screwing up their team formation and it will also prevent the enemy from getting away from the damage by preventing them from flashing away and making them become easier targets for skillshots.
Damage:
Generally casters are a lot weaker at level 1 because they are limited to one skill and their base attack damage and attack speed are very low! This will lower the amount of damage they can deal.
When choosing a strong early game damage dealer, take a look at their base attack damage and their skills. A champion like Ganglank, has a strong early game because he has multiple sources of damage. His auto attacks can proc his passive that can not only deal additional damage but provide a slow, AND his Parrrley which has a relatively low cooldown can add a lot of additional damage.
Crowd Control:
Generally a team that is able to stop a enemy from running away or actually brings the enemy to the team will have the upper hand in team fights. Morgana's Dark Binding starts out with a 2 second snare. That is quite a bit of time to stand still. Blitz who takes his grab at level 1 will literally serve one of the enemies on a platter if one of them is pulled into your team.
Passives:
Not only do we have to look at the skills a champions have but also their passives. Some champions passives will be way more useful in level 1 fights than others. For example, Volibears passive will let him regenerate 30% of his health back when he reaches a certain threshold on his health bar. Or Anivia who turns into an egg when her health bar reaches 0. Contrast that to Ryze who gets a 1 second cool down reduction for each spell he casts. But that reduction can't be utilized at all since he can only use one skill at a time. Or Ahri who gains life steal for one attack after hitting 9 targets. This is great in all for sustain but during the course of a team fight, the passive will not really play a big factor.
Non-target skillshots:
Sometimes in a level 1 fight, you get to an impasse where neither teams are willing to iniate and are just standing just out of range. Having a champion with a long non-target skillshot harass will generally mean an easier team fight OR allow you to push them back if you are indeed trying to invade their jungle. These skills include Nidalee's spears, Lux's Lucent Singularity(which not only damages but also provide vision for 5 seconds if she does not detonate), and Ezreal's Mystic shot.
Vision
If there is a support on your team, ask them to take Clairvoyance and to buy wards(technically, you shouldn't need to ask them if they are playing support). Vision is very important especially in early games because a champion can be focus down very fast if you can see them. If you are invading their blue, or if your blue is being invade, throw a ward onto the bush right next to where the golem spawns. Once the team fight starts, the support should throw the Clairvoyance right behind where the enemy is so provide vision of all of them. Vision in the jungle is very limited because, the bushes and the corners in the jungle will inhibit vision.
Initiation and Team Formation
Because in level 1 fights the tanks have not begun building tanky items yet, there really isn't a tank per se. Anyone who runs into the fray can easily be focused down in seconds. Because of this, it is better to stay in formation (champions with higher defensive base stats in the front, and champions with lower defensive base stats in the back), and play reactively. If one of them charges in, then focus that champion down. Generally it is smart to have your "formation" near a bush so the squishy targets can utilize that to avoid being a initial target in the fight.
Items
Since Champions start out with 475 gold or 515 gold(if you put the two points in the ultility tree), items do not play as big of a factor. However generally buying Dorian items will give you more bang for your limited buck and give you an early game advantage.
in the ggClassic Epik vs CLG game, CLG ran a "kite" comp.
They basically ran fast move speed champs that had very good poke and would kite epik until they were low enough to collapse on. It worked so well it was scary.
I have also seen true damage comps, with vayne, olaf, cho. It is kind of one trick but the focus target is guaranteed to die.
For a college tournament, I had a few comps pre-planned based on certain team synergy tricks, not really high elo or anything more theorycraft on beastly mechanics. We ended up using a couple:
Super stun gank heavy - Sion, taric, ashe, veigar, udyr
suppress heavy - malz, ww, scarner, taric, trist
Best Answer
EDIT: This is really out-of-date, and I'm working on re-writing it. I don't recommend following the advice given here just yet.
In most cases, putting all your eggs in one basket by choosing champions that all fit the same role as tanks, healers, dps, or some other mechanic will not result in a good team. A good team has champions that serve different roles and have lots of synergy with their teammates.
The most important role is a tank. Without a tank it's difficult to initiate team fights and you team will lack someone to prevent speedy gankers like Master Yi and Evelynn from killing weaker teammates. Champions that fit this role have to be able to soak a lot of damage and provide crowd control in order to protect teammates. The most useful crowd control for a tank is a taunt, which will pull your opponents attention away from a teammate they are trying to take down. For this role you want champions like Rammus and Shen, although there are many other viable tanks.
Support is also very important. Support is a rather vague word but it usually means healing and crowd control. This lets support step in to help the tank keep other teammates alive, as well as serving dps when the support isn't needed. Janna is widely played support due to her AoE heal, her shield, and her two crowd control tornadoes.
Beyond those two roles your champions will be focused on dps. However, there are still some things you should balance.
Magic Damage and Physical Damage: If you don't balance the two then your opponents will stack resistance for whatever damage your team deals. An example of this is a physical damage team being entirely useless against Rammus, who gets up to 75% physical resistance and deals damage back each time you attack him. Choosing a mage like Vlad and a fighter like Ashe will be enough to balance your damage.
Crowd Control and Nuking: Crowd control keeps your targets from running, and nuking does high damage. Choosing only one or the other means you will see targets escaping easily or not taking enough damage. An example of a champion who balances the two is Morgana with Dark Binding, which nukes and roots.
Damage Focus and Damage Resistance: Champions that focus on damage tend to lack in resistance and health. If all of your dps champions are squishy, they will be targeted in team fights and likely die, leaving your team with no damage output. Balancing damage and resistance tends to be in choice of items more than champions, but choosing some tanky dps like Sion and Jax will help. Also, tanky dps can step in as a temporary tank if your team tank happens to die.
Picking champions with AoE and global abilities is not necessary. These spells are scaled to be roughly as powerful as single target spells and thus provide marginally more value in team fights, and no more value in situations where there is only one target (i.e., you are ganking or getting ganked). AoE is only super advantageous if your team is having difficulty focusing one target.
I haven't mentioned much about synergy but it mostly comes from your team's capabilities. Do you have a nuke to follow up a nicely placed stun? Can your tank be helped when he gets too deep into a group of enemies? Think of different situations and whether you are capable of coming out on top of them as a whole or only a small part. And if you want an idea of team fighting synergy (this largely promotes AoE) watch the circle of death.
Example Team
This is what I consider a good team, based on my points above:
The Roles:
The only champion that doesn't entirely follow what I recommended is Galio. However, Galio was chosen because he can deal lots of dps with enough AP focus, but he also protects teammates with Bulwark and has a Taunt to help him sub in as a tank. This means Galio can be another tank, dps, or support with enough focus. His versatility makes him an easy choice for me.
One weakpoint is that the team lacks healing, but Galio, Nunu, and Morgana have self heals (Morgana has spell vamp) and Jax usually takes some vampiric item later game. Rammus is a tank so choosing a health regen item for him like Force of Nature or Warmog's Armor can be enough.
The Synergy
Other than the roles they fit these champions also have a lot of synergy, mainly in their ultimates. Rammus can initiate a large team fight, which is when Galio and Morgana step in. Morgana casts her ultimate, and Galio taunts to keep enemies in. Then Nunu steps in to cast his ultimate, and Rammus and Jax use their cc to keep opponents close to Nunu. The sheer dps is enough to completely kill or scatter a team, and despite exhausting most of your ultimates your team still has a lot of power as each champion has cc and dps.