I maxed out at 305 lbs when I was 19. I am now 23 and am 280

Without knowing how tall you were, I estimated your TDEE at 2474 Calories. Keep in mind that as you go down in weight your TDEE will drop as well and you'll need to eat less to maintain a deficit.

goal is to lose 50 lbs by the wedding [in October].

Generously, lets say that is 4 months away (bearing in mind that July is 1/2 over and that this calculation goes through the end of October instead of whatever the date in October this wedding is).

4 months x 4 weeks = 16 weeks deadline, so you should realistically look at losing about 16 pounds by then instead of 50, especially if you're eating at 1900-2000 calories per day as you said in you comment. That's only roughly a 500 calorie deficit which would drop you about 1 lb per week in the best conditions.

My predicament is that I don't know what to do in order to lose weight

Weight loss is 100% about creating a calorie deficit. Whether you do this by burning off calories through workouts or by simply not eating those calories is up to you, but it's far easier to just not consume them than try to burn them off.

I'd recommend going with a 1500 calorie daily budget. This would put you at about 900 calorie daily deficit which would burn you almost 29 lbs by your 16 week deadline. Because your TDEE will drop down as you progress though you may only lose 25 or so, but that's still a good start!

Do any of you know any exercises that I could do while sitting at a chair all day?

I get bored of sitting all day so I tend to squat for a while, do dips at my cube, plank, etc, but none of these are realistically going to burn enough calories to cause a noticeable difference.