EDIT: This is really out-of-date, and I'm working on re-writing it.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.
This is what I consider a good team, based on my points above:
- Rammus (our team tank)
- Galio (our support, for the most part)
- Morgana (our mage)
- Jax (our fighter and tanky dps)
- Nunu (more magic dps and crowd control)
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.
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.