Flying was the only holdout with no pure type pokemon, which as you note Gen V changes. From Bulbapedia:

Prior to Generation V, this type was the only one of the seventeen that was never seen as an authentic pure type; Tornadus is the first pure Flying type.

There are, however, a large number of type combinations that don't exist.

It's not about the individual

As was said in Williham Totland's answer, it's about the overall typing of the team, not the typing of the individual pokemon. No one pokemon will be able to rise to any challenge you face, but that is why you have room to carry six with you.

A good starting point is to pick pokemon with different typings, both offensively and defensively. Let's look at an example of what I mean by this.

Example Pokemon

Defensive Analysis

Blastoise is a Water pokemon, so he's going to be strong against fire, ice, steel, and water attacks, but weak to electric and grass attacks. It may look like Blastoise is a good choice because he's strong against more things than he's weak against, but if you look more closely, it actually comes out pretty evenly. Electric and Grass are two very prevalent types of moves, especially grass. You can expect almost every trainer and grass patch in the first half of the game to have a grass move of some kind. Conversely, being strong against steel isn't that big a deal because those moves are very rare and the pokemon who use them generally don't have a very strong attack power.

Offensive Analysis

Offensively, Blastoise gives access to STAB surf, which is extremely strong, as well as some ice attacks if you use a TM. He really doesn't have too much versatility at all.

Example Team

Those are the kinds of things you should take into account when looking at an individual Pokemon, but what about a team? Let's say you have a Venusaur, Golem, Pidgeot, and Dragonite. There are a lot of different types represented here, so at first glance it may look like it's well-rounded. The problem, however, is the glaring weakness to Ice. Venusaur, Golem, Pidgeot, and Dragonite are all weak to ice. If you don't do something about this, you're probably going to have trouble taking on any kind of ice trainers once you get later on. You may want to consider adding in Rapidash or Primeape.

General Strategy

The general principle here is to take a look at your team as a whole. You want to try to use pokemon who have a wide array of offensive options and few defensive drawbacks. You also want to make sure that, out of your six, you have a few ways to deal with common threats and at least one way to deal with any kind of threat. It's okay if you don't have a fire attack on your team, as long as you have a flying attack and a fighting attack to cover the cases where fire would have been usefel. (Fire is strong against grass, ice, and bug, so having flying (strong against grass/bug) and fighting (strong against ice) will cover the loss of a fire move.)

Generally Good Types

Particularly strong offensive types for playing through casually are:

• flying
• fighting
• ground
• rock

Particularly strong defensive types for playing through casually are:

• normal
• rock
• steel

(Note that all of these defensive types are weak to fighting, so make sure to include something to compensate for that.)

One Last Bit

The best advice I can give, to wrap it all up, is to pay attention to what you have difficulty with, and adjust to compensate for that. If you are consistently having trouble taking down electric pokemon, add a ground pokemon to your lineup. If your Jigglypuff never contributes anything positive to the fight and just comes in to give you a turn to heal up someone else, maybe consider replacing it with a better alternative.

_{Sorry for using all first gen Pokemon in my examples. I don't know fifth gen as well off the top of my head, and I think this may make the answer more accessible to more people anyway}