All 忘れた means is 'I forgot'. It's a statement and doesn't carry any additional meaning besides the fact that somebody forgot something.

On the other hand, 忘れちゃった is a colloquial, shortened version of 忘れてしまった.

～てしまう is a verb-only conjugation and has three basic meanings, which are:

1. The action expressed by the verb was done unintentionally, by accident.
2. The action was done completely, finalized.

In this case, 忘れちゃった may mean both, as in:

Whoops, I totally forgot.

忘れてた means 'I forgot', but it carries a sort of 'continuous' meaning, i.e. somebody has forgotten something for a period of time. At least that's how I've come to understand it.

On the other hand, 忘れちゃった is a colloquial, shortened version of 忘れてしまった.

～てしまう is a verb-only conjugation and has three basic meanings, which are:

1. The action expressed by the verb was done unintentionally, by accident.
2. The action was done completely, finalized.

In this case, 忘れちゃった may mean both, as in:

Whoops, I totally forgot.