Tap the event. It should pop up with some more info, and an Edit button. Tap the Edit button reveal all the event's settings and make changes, or Delete. The Delete button is all the way at the bottom of the Edit popup (scroll down).
Depending on how you added them, not all calendars may be editable.
I know to hit delete if it is given as a choice How do I delete an event on my calendar if all it says is details.
If delete is not a choice, then you are probably trying to delete events from a calendar that you subscribed to, rather than sync with. Subscribed calendars are usually one way. You have to delete the events from the other end.
Calendars that were subscribed to should show up in a special account called Subscribed Calendars (if memory servers). You'll find them in the same place as email accounts; at the top of Settings > Mail, Contacts, Calendars. It is also possible that some of the calendars you are synced to are read only on your server side as well; that's possible with GMail (and probably MS Exchange) at any rate.
Without knowing more about how you've set up your calendars, what you are syncing with, etc. I can't make any better guesses. Even then, I don't have an Exchange account to play with so I can't give any solid advice on read/write calendar sharing there.