Desculpa and desculpe mean “I’m sorry” in the sense of “I apologize”. The former is casual, while the latter is formal. Otherwise, they’re the same. You can use them when you bump into people, when you do anything wrong… Any situation where you feel like you might be at fault.
Lamento and Sinto muito mean “I’m sorry” in the sense of “I’m sorry to hear that”. You’d use it to console someone who just got some bad news, for example. If the news involve someone passing away, you could also say/add Os meus pêsames or As minhas condolências (both mean “My condolences”).
These are all used a lot in their respective contexts, but to apologize, you do have an extra option: Perdão. Interestingly enough, when people cough or burp (ha!), this is what they’ll say the most. Otherwise, it’s not the most common way of apologizing.
UPDATE - Here’s a Learning Note on the topic Com Licença? Desculpe? Perdão?