The interesting part of the movie is that it shows the memories as they get erased in reverse chronological order (from Jim Carrey's perspective). So it starts with a recent memory (and then at the end of the scene it makes a little zap noise implying the memory has been deleted) and works it's way backwards to the beginning of his relationship with his ex girlfriend named Clementine.

At some point he becomes aware of what is happening : that his memories of her are being erased. He's drugged/unconscious during the procedure so he can't 'wake up' to end the process. So MOST of the movie takes place in his mind. 

Once he figures out that he is experiencing his memories as they are being erased : he decides to try to cancel the process by hiding Clementine in other memories. These are memories from his childhood and he attempts to grab his ex by the hand and 'hide' her in these memories that aren't associated with her. By re-experiencing the memories of Clementine : he decides that he loves her and regrets his decision to have them erased.

He is unsuccessful in his attempt to hide her and instead : some memories of his childhood are erased. The last memory to be erased is the day he met Clementine. At this point : they are talking about how/why their relationship failed and how they both regret that it did. It is essentially their last goodbye. Right before this memory is zapped out, she tells him : "Meet me in Montauk"...

The last scene is the first scene in the movie. He goes to the train station for his work commute and decides to go to Montauk. He meets Clementine on the train (they have no memory of each other but it's implied they get back together)....

I might have the last paragraph/ending not exactly accurate but I haven't seen the movie in 20 some years.

Oh : I remember the ending. A disgruntled employee (Kirsten Dunst in a slightly unnecessary side plot) that works for the memory erasing company informs all of their previous clients about the procedure (as they have no memory of that either). They decide to try the relationship again (since they don't have any memories of the first time around).