When you fill out your graduate school application, you will supply the e-mail of the professors that you are getting LORs from. Your professor will get a LOR request in their inbox and go from there.
Along with a letter, some schools also require a supplemental form for the professor to fill out. Two of the schools I applied to required one. They had my professors rate me on a scale from 1-5 on things like "Chance of being a successful SLP", "Writing Skills", and "Ability to Verbally Express Ideas". They will send this form to your professor.
When you fill out your application there is a box that says, "I waive my right to see my LOR". You can find a lot of info. online about it. I like this write up: http://wiselikeus.com/collegewise/2010/09/should-you-waive-your-rights-to-see-your-letters-of-rec.html
Like Jay said, make sure you ask your professors well in advance. Some professors have to write A LOT of rec. letters. Also, my professors knew me very well, but still asked for me to bring them my transcripts and my resume (I even gave one of my professors a copy of one of my research papers). It couldn't hurt to have that information ready to give them when you go in to talk to them about writing a letter.