My panel box and breakers are Bryant, with about 20 positions for breakers. My question is, my main breaker is 2, 60 Amp breakers doubled, What is my service? 60, 120 ? And what are the total amount of 15 and 20 amp breakers I should have in this panel?

If you have breakers you have bigger than a 60A service.

Does the 60 say "lighting main", or something like that?

How old is this panel?
What other breakers are there?
Can you post a pic of the panel?

The breaker, and service is rated 60 amps 120/240 volt. The total amount and rating of the branch circuit breakers do not mean anything relating the main breaker or service rating.

The service is sized according to several load calculations based on the size of the home, and several mandatory circuits, and the optional appliances there maybe. I will not show laypeople how to do the calcs, due to various variables, and just plain lazy.

Where do you need to go with this?