Dr. Richard Burke was Monica's most serious relationship prior to being with Chandler. Richard is said to be Jack and Judy Geller's best friend, and Monica was friends with Richard's daughter Michelle in high school. So there was quite a family history before they got together.
Moving on... Richard is an Ophthalmologist by trade, is 21 years Monica's senior and fairly recently divorced from his high school sweetheart, Barbara. He is the father of two grown children: Timothy (also an eye doctor) and the aforementioned Michelle. He also has two grandchildren via Michelle.
Monica and Richard end up (somewhat reluctantly) revealing their budding romance to Jack and Judy Geller at Jack's birthday party. Monica didn't want to do that, because she didn't want to give her dad a stroke on his birthday. However, people are gossping about Richard's new "twinkie in the city" and they feel they have to tell. Monica's parents are initially stunned, but their feelings about the relationship aren't really discussed further.
Monica and Richard have a functional relationship, relatively free from conflict. However, Monica wants children and Richard has already raised his and doesn't want more. Monica tries to give up on the idea of having children, but she can't, so they part ways (02.24). They attempt to be "friends with benefits" several months later (03.13), but end up right back at the same crossroads and call it quits again.
Later on in the series, Monica actually starts to date Richard's son Timothy; however, she soon discovers he kisses like Richard. It freaks them both out and they end things, basically as soon as they began (04.08).
When Chandler is preparing to propose to Monica at dinner, the two run into Richard and a date. The encounter stalls Chandler's plans, and prompts Richard to realize he still loves Monica. Richard visits her at work and declares he's ready to give Monica all she wants (marriage and children). Meanwhile, Chandler, afraid Monica suspects he's going to propose, starts to act as though he wants nothing to do with marriage (so that it will be a surprise).
Monica visits Richard and tells him she isn't sure what to do, but that what he's offering is what she wants, not the immature conversations she's been having with Chandler of late (as part of the throwing her off course plan). Chandler arrives at Richard's, expecting to find Monica, but she's already gone. He confronts Richard about declaring his love for Monica and offering to marry her. He explains that he was just pretending to not want marriage, so that she'd be surprised. Richard replies, "if it helps, it worked very well." He tells Chandler to go get her and never let her go. Chandler tells Richard he is a good guy, to which Richard says, "I know. I hate that" (O6.24-25).
Richard is never seen in the series again after this point, though Chandler is still slightly insecure when it comes to the very mention of him.