I can't give you a rundown of every single character, but I'll pick from a couple that I've played quite a bit.
Lucio (Support)- Fast-moving healer who lets you have fun on the offensive side as well. He can toggle his AOE buff from a speed boost to a passive heal that affects all allies within a certain radius. He can further boost this every 10-12 seconds (can't remember the exact cooldown) for an added bonus to movement speed or heal rate. Some players can find healers to be boring, but Lucio keeps your attention because his attack isn't entirely useless and his alternate fire (right click) acts as a shockwave that pushes nearby enemies away. This is especially fun (and highlight reel material) as you can launch enemies off of edges on the map to their death. He is especially effective on King of the Hill formats where a lot of allies are clustered in close proximity, allowing him to heal several teammates at once. His ultimate provides a massive AOE buff that, when correctly timed, can win a team fight.
Widowmaker (Defense/Sniper)- Mobile sniper. She uses a grappling hook to get around and her headshots are deadly for all non-tank characters. Her ultimate provides vision on the enemy team (basically wall hacks). She also can throw down a poisonous mine that causes damage over time. She is a pretty straight forward character, so someone coming from a more traditional FPS background should be able to get used to her pretty quickly. Her sniper rifle functions as an assault rifle when not scoped as well making her not entirely useless in close range, though you still want to avoid those engagements.
Roadhog (Tank)- Tanky with a punch. Roadhog is frustrating to play against when controlled by someone with skill. He can hook opposing characters (similar to Pudge from Dota) and then finish off squishy characters with a quick shotgun blast to the face. The hook is something that takes some getting used to, but when mastered he is a very effective character. He works well in close with his gun, but his alternate fire is also fairly decent for whittling away the defenses of other characters from a distance. He can also heal himself for a large portion of his overall health with an ability every 8 seconds (though he is immobile and unable of any other action while this ability is being used). His ultimate is a machine gun that deals decent damage and knocks back enemies that are hit.
Tracer (Attack)- Though I'm pretty useless as Tracer, she is a very fun and unique hero to play when picking up Overwatch. She can dart (or blink) in any direction that she is facing (holds up to 3 charges and recharges fairly quickly). Her second ability essentially "rewinds" time and recovers any damage that she took. So, if Tracer finds herself in a jam she can essentially rewind the damage dealt and also return to the position on the map that she was before she got into trouble in the first place. Her ultimate (sticky grenade) is somewhat weaker than others, but if used correctly (in combination with her blink or rewind ability) it can do quite a bit of damage. She attacks with dual pistols that hold small clips, but when up close and pumping a full clip into the enemy, she can make short work of pretty much anyone.
Those are four different characters from four different roles in the game. Hope that gives you some idea of these characters meanwhile I'm sure others will be able to help you out with the rest.
I've had quite a bit of fun with Overwatch so far, and I played during the Open Beta as well. I have the most fun when playing with a group of friends though as it is a very teamwork heavy game so communication is always a benefit, especially when coordinating team compositions. So, if you know others that are playing, that would make it all the better. That being said, it is still a fun game running solo, and when you play well it's easy to get other players to add you for a regular group to queue up together.
