To register your foal, you will either need to get the current owner of record to sign her over to you so you can transfer her papers into your name, or else the current owner of record would need to sign all the paperwork to regiter the foal and also sign a transfer of ownership for the foal (infinitely simpler to just transfer the mare so that you can do all of the foal paperwork yourself.)
What breed and registry is she registered with? Do you have her papers? Do you have a signed transfer and/or is the back of her papers signed (for Arabians and TBs)?
OK. For AQHA, you need her original registration certificate, and you need to check and see who is recorded on it as the most recent current owner. Then you need a transfer form signed by that person as the seller.
Seller needs to sign it, and you need to fill it out with you listed as the buyer.
Then send in the original certificate, the signed transfer, and all fees (transfer fee and membership fee if you are not already an AQHA member) in to AQHA, and AQHA will send back the registration certificate with you listed as the current owner, and you will be listed as the current owner in their records as well.
Make sure the date of sale you list on the transfer of ownership BEFORE the dates you start breeding the mare, so that you are her recorded owner at breeding and foaling time-- that way as the mare owner you can sign the relevant parts of the registration application for your foal.
If you don't have the original certificate, but know the horse's registered name and/or number, you can call AQHA to find out who the current listed owner is, and the current owner would have to sign an affidavit to get a duplicate registration certificate issued, in addition to the signed transfer. http://www.aqha.com/forms/pdf/AffidavitDupCertWEB.pdf