You get free live membership for like a month when you buy your xbox. Other than that I haven't heard of a free live membership.
If you were on live before and now you can't get on all of a sudden, I would suggest renewing your membership which would cost money. Most likely right now you have a silver membership which is kinda deceiving at first because its like a gold membership. You can still see all your friends and what not but you can't talk to them (I believe) and you can't play games through xbox live.
Without further description of the problem though it's kind of hard to pin point exactly what's going wrong.