The term "Well-Rounded" has so many factors that both casual and competitive players keep in mind when choosing their Pokemon, but most importantly, picking out THEIR TEAM.
First off:
Casual Gamers
Weaknesses:
If you're considering a "Well-Rounded" approach, consider using Pokemon that have few or no notable weaknesses. Pokemon like Spiritomb, or Eelektross* are among a few Pokemon that have no known weaknesses (meaning, no super-effective attacks on them).
As other people have mentioned above, try picking Pokemon that are multi-typed, meaning, they carry more than one "type". Having more than one type usually opens up more movesets for your Pokemon, allowing you to fit a lot more different and powerful moves into your team. For example, having a Thunder/Water Pokemon is nice because, although Pokemon with only the Water type would take super-effective damage from Thunder-types, a multi-type does not face such problems, and helps to cover any gaps in a Pokemon's abilities.
Ability types:
You'll also want to consider different "ability" types. Certain Pokemon have better ability types than others, some good, some not so good. For example, the pokemon Shedinja has the ability Wonder Guard, which nullifies all damage that isn't a super-effective attack. Note that Shedinja can only have a maximum of 1 hp, but that means the only attacks that can actually kill it are Dark, Fire, Flying, Ghost, and Rock, which means you have to only worry about Pokemon with those type attacks.
Another example is Yanma, which one of his abilities, Compound Eyes, grants a 1.3x bonus to accuracy. This can help fill in some moves in the game, for example, if you have a move that doesn't have a 100 accuracy (meaning it may miss), Compound Eyes increases it to the point where it's almost a guaranteed hit. So consider having different Pokemon with different abilities; there are a TON out there to choose from.
Now since you're playing casual, and not getting into the messy world of Competitive players, you might want to consider getting an HM mule. Since most casual players spend their time travelling throughout Unova exploring places, it might be a good idea to carry Pokemon with abilities that can be used both on and off the map.
Mules:
A good example is Bibarrel, a more famous pack mule. Because HM's are so hard to get rid of (once you progress to the end of the game, there should be a HM remover trainer), and usually not that strong, it may be more effective to give HMs to another Pokemon in that party to take care of. As I mentioned, Bibarrel is capable of taking the HMs Cut, Surf, Strength, Waterfall, Dive. which helps to cover a lot of the travelling you'll do throughout Unova. Another example is Pelipper, which can learn the mvoes Fly and Surf, two of the most important moves in game.
Variety
You'll definitely want to have a multitude of different Pokemon in your team. Say for example you go up against a Fire type Pokemon, and unfortunately, your whole team is full of Sunkern. Obviously, you're going to be in for a huge thrashing, that Fire type will raze through your whole team. Now say we had the same situation, but you had 5 sunkerns (not much better), and 1 water type. This means you actually stand a chance, and can possible take down that Fire type pokemon. Basically, don't just center your team around one type of Pokemon, or one type, try mixing it up, and trying to vary it a bit. Having extra Pokemon that fit certain roles is always good; it's pretty much a good way to switch up your teams.
STAB
STAB stands for Same-Type Attack Bonus. Basically, let's say you're a water Pokemon, and you have a water type move, Surf. Surf has a base power of 95. What STAB basically does is, if your Pokemon has the same type as the type of the attack, it adds 50% bonus damage to your attack. Meaning, you should match moves of the same type as your Pokemon, and the same for multi-type Pokemon. As mentioned before, the Pokemon Pelliper has a typing of Water/Flying. If you teach it Fly, and Surf, that means you get a damage bonus whenever you use Fly, because it's a STAB bonus, and the same for SURF. Keep this in mind to make the most out of your attacks.
TL;DR Casual Version Pick Pokemon with multiple types; this will allow a broad range of moves and different abilities. Having different Pokemon in your team, as well as in your PC, works wonders, meaning you can substitute what gaps you need to fill any time.
Competitive Gamers
I'm not much of a Competitive player; the scene there is a lot of experienced, well-polished players who could probably take me down any time. In a competitive setting, there is a bigger emphasis on the team aspect, meaning, all the aspects of the team should synergize well enough with another.
Teams:
When playing competitively, there is a huge emphasis on crafting the right team with the right Pokemon with the right EVs and IVs (i'll get to it in a bit), with the right abilities, with the right moves, all with the right items along with the right nature. Yeah, that's quite a mouthful. Basically, every single detail you probably wouldn't care about in a Casual perspective is taken in, magnified to the maximum detail, and carefully analyzed to produce the best team. Since there are fairly notable Pokemon to watch out in competitive matches, meaning, Pokemon that are fairly commonplace on teams because they are THAT good.
Movesets:
This is where Competitive players may consider using moves which have high attack levels (usually at the cost of nerfing some stats), priority moves (think Quick Attack), or even Buff Moves (think Defend (raises Defense)). Basically, it goes down to a whole huge world of detail I'm not too experienced with.
EVs and IVs:
I'm not too good with this, I'll just leave a really comprehensive link here, as I don't want to take their credit. Basically, the jist of the matter is, when you defeat certain Pokemon, you end up gaining a certain amount of "Effort Value (EV)" points that contribute towards certain statistics.
TL;DR Competitive Version It takes a lot of time planning, making teams, setting up teams, and considering each battle turn by turn to get really good at this.
Other Notable PokeLinks:
Some good websites to frequent if you ever get into Competitive gaming, or even Casual gaming are Smogon (competitive), Veekun (virtual Pokedex), Serebii, Bulbapedia.
*Though Eelektross does have a weakness to Ground based attacks, his ability Levitate completely nullifies incoming Ground type damage, meaning this Pokemon has no virtual weakness.