Roaming has become a popular addition to the metagame strategy, wherein one player does not lane nor jungle, but rather simply moves through the jungle/river to initiate ganks on lanes from level 1 and onward, until the teamfight portion of the game begins.
A roamer essentially guarantees that you're either running a solo lane, or two solo lanes if you have a roamer and a jungler. I've also seen a double roamer game played, though this tends to be less effective as enemies who are wise start playing defensively and reduce the effectiveness of your roamers, which put them behind in both CS and levels.
The qualities of a roamer can vary, but the general requirements are that they are mobile, can do burst damage, and have a strong stun or position altering (push, throw, etc.) ability. Taric, Evelynn and Alistar are all very popular choices for roaming. I've also seen Sion and Blitz play roaming effectively, but they were both out of a lack of a jungle spot on the team and normally suit better in a lane.
This strategy is similar to the "double jungle" mindset in that you get three solo lanes, with the exception that there is extra pressure on getting early kills to succeed. A definite risk vs. reward tossup, as successful roamers will quickly catapult your laners ahead in level / CS as well as demoralize your opponents, but unsuccessful roamers will fall just as far behind due to the lack of their presence in the laning phase.
Squishiness is essentially how easily you fall during an engagement - the relative ease of which you can be squished. Whereas a tanky champion can be likened to the solidness of stone, a squishy one has the standing power of a pillow. It is one of the many factors of general survivability - the strict ability to withstand damage.
Generally, squishiness manifests in champions with low defensive stats, little or no defensive ability to prevent death under focus fire from the enemy team, and poor ability to trade blows. The general theme is, if you are squishy, your likelihood of making it through an ambush, or under fire in a team fight, is low to nonexistent.
You might notice that a lot of these are matters of "lacking". And that's what squishiness boils down to - lacking defenses. As it is easier to illustrate that way, example champions I'll list will generally be to show abilities that reduce squishiness - squishy champions thus being those who lack such abilities.
Low Defense Stats
Health, Armor, and Magic Resistance all play a vital role in not dying. Whereas a tank will excel in these stats (and further boost these with items), a squishy champion tends to have poor base stats and/or poor stat progression per level. Without even the weakest of defensive items, a squishy champion will fall to pieces when under attack.
Vladimir is an example of a champion with reduced squishiness in this department. His passive grants him extra AP for bonus health, and extra health for bonus AP. This encourages him to buy health items, while at the same time benefiting his health anyway while he builds offensively.
No Personal Death Prevention
Naturally, any champion, even a tank, will fall under enough focus fire, but squishies are those who die within moments of focus. This happens both from a lack of defense stats as before, but also from lacking abilities that help mitigate damage. This includes damage reduction, damage shields, or any method of instant damage negation - things that help once the damage actually starts to come in.
Poppy is an example of a champion with reduced squishiness in this department. Her passive sharply reduces damage, she gets stacks of Armor, and her ultimate lets her negate all damage except from one source. This both lets her survive once the damage starts, and even completely negate the effects of focus fire.
Poor Trading Blows
Likewise, being able to simply kill an enemy before they can hurt you doesn't affect your squishiness. That ability won't help you if you get jumped from the brush, or if you are on cooldown and took a wrong turn. If an initial burst doesn't take down the opponent, then a squishy hero may not survive the continued exchange of attacks with a foe, even if initially at a health advantage. Again, defensive stats play a large role in this, but also abilities to help you stick around in the fight without needing to retreat.
Graves is an example with reduced squishiness in this department. His passive increases his Armor and Magic Resistance, so he gets a growing advantage if he survives an ambush. As the rest of his kit also supports trading blows on the offensive end, his reduced squishiness really improves his survivability in that department.
Fixing Squishiness
Now that you see what factors (or rather, lack of factors) play a role in being known as “squishy”, what can you do about it?
The main things to look at are fixing your squishiness directly with items, or just mitigating its issues. Fixing it directly is basically - buy defensive items! Every little bit of health, armor, and magic resistance goes a long way if you’re a squishy champion. Many high tier items include those that give some measure of defenses in addition to offensive power (Abyssal Sceptre, Frozen Mallet, Wriggle’s Lantern). You may also consider items that directly present potent defenses - Banshee’s Veil, Hexdrinker, and Zhonya’s Hourglass all offer a variety of ways to handle your squishiness.
Mitigating it comes from equipment that helps your survivability otherwise. Lifesteal/spell vamp, for example, won’t save you from a nuke barrage, but it will improve your ability to trade blows by healing you for each hit (a fact which for AD champions is very nicely wrapped with Wriggle’s Lantern). Also consider items like Randuin’s Omen, which can slow down the enemy’s attack speed so that they cut through your buttery softness more like a dull knife than a laser beam.
A champion with a stun, silence, or snare can also use those abilities to stave off the flow of damage - if you don’t have one of your own, lane with someone who does. In fact, laning with someone who can protect you either with their own defensive shields or by holding off the enemy is a generally wise move.
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Edit: Thanks! I tried to improve this answer with a few more examples. Don't hesitate to give feedback.
Rule #1: Player skills over champion matchup. Don't pick a champion you don't feel confident with, just because it should counter your opponent. (Unless you want to practice, of course.)
Rule #2: Don't blindly trust LoLcounter. Why should you?
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That being said... There are indeed game mechanics that favor some champions in a given matchup. A few examples:
Ryze can root (W, "Arcane Prison") an opponent before damaging him. Kennen's "woohoo" sprint (E, "Lightning Dash") makes him unable to do anything except moving fast, and that is his main farming tool. Therefore, farming with Lightning Dash in front of Ryze is almost impossible.
Short-ranged champions with no gap-closing abilities, e.g. Udyr or Ryze (yes, he has ranged attacks, but his range is shorter than most AP carries) are weak to kiting. Ashe counters Udyr (he can never reach her) and Cassiopeia counters Ryze (she can harass him with Q forever while never entering his casting range). Among AD carries, Caitlyn counters Vayne for the same reason, even though it is less obvious.
Some champions have silencing abilities. They tend to fare better against casters, who rely on casting spells to deal damage.
Jax can dodge basic attacks during his majorette stance (E, "Counter Strike"). That gives him an edge over champions who mostly rely on basic attacks, e.g. AD carries, Tryndamere, AD Master Yi, etc.
Lee Sin has two abilities revealing stealthed champions (Q, "Sonic Wave", and E, "Tempest"), i.e. he counters Wukong, Akali, etc.
Kennen's and Fiddlesticks' ultimates ("Slicing Maelstorm" and "Crowstorm") are area spells that follow them and deal damage over time around them. Which means they will be wasted if the caster is blown away from the fight. Janna's ultimate ("Monsoon") is perfect for this; she is a famous counter to Kennen and Fiddlesticks in team fights only. Other champions with knock-back abilities: Alistar, Jayce, Gragas, Lee Sin and Tristana.
Katarina's ultimate ("Death Lotus") deals huge damage over 2 seconds around her. However, this is a channeled ability. Any crowd-control ability will stop it. Three out of Blitzcrank's four abilities can interrupt Death Lotus, especially his ultimate ("Static Field"), which is an instant and very safe way to disable Katarina. She cannot hope to land her ultimate in the middle of Blitzcrank's team.
Poppy's passive ability ("Valiant Fighter") reduces by half the damage taken from heavy hits (more than 10% of her current health). It makes her highly resilient to spike damage, such as Veigar's spells. However, she is very weak to damage-over-time spells (which are almost never affected) and true damage (which cannot be mitigated). Teemo, Darius and Olaf counter Poppy in top lane.
Some of Malphite's and Rammus' abilities benefit from armor ("Spiked Shell", "Defensive Ball Curl" and "Ground Slam"). They are more efficient against teams that rely a lot on physical damage. On the other hand, 50% of Galio's magic resistance is added to his AP ("Runic Skin"); therefore, he is better against teams that rely on magic damage.
"Yorick walks into a bar... There is no counter." (This is wrong, yet almost true, but I love this joke. Cho'Gath, Nasus and Trundle may benefit from Yorick's ghouls.)
These are only examples. They might even become outdated someday. However, what truly matters is the understanding of game mechanics that leads to these examples.
Always ask yourself "what makes this champion strong? what would stop it in its tracks?".
Bottom line: be your own LoLcounter.