One day they were the best of friends. The next morning they were even closer...
Chandler and Monica had been friends for years. Chandler was Monica's brother's best friend in college and beyond, and they had always hung out together. In fact they lived Across the Hall from each other.
Chandler had always been there for Monica, whether it was comforting her in the waiting room at the hospital when she was depressed about never having a baby. It was then that Chandler offered to get together with her if neither of them were married by the time they were 40, thus becoming each other's 'backups'. Well Monica actually took offence, but in the hearts of the Chandler and Monica fans, the deal still existed.
In the years leading up to their relationship there were never any major hints about and attraction, but they were always hugging each other, comforting each other, and teasing each other.
Then on a trip to London (baby!), a drunken man mistakes Monica for Ross's mother. This sends her in a downward spiral to alcohol heaven, when Chandler comforts her. The next time we see them:

During the next year they try to keep their relationship a secret, but when everyone finds out they handle it well, because they trust each other. On their first anniversary, they take a trip to Vegas, where the roll of two fours inspires Chandler and Monica to find the nearest wedding chapel and tie the knot. This is soon forgotten when Ross and Rachel stumble out, but luckily the relationship does move forward as, back in New York, Chandler asks to move in with her. She accepts, and the crowd goes wild.
They spend their time adjusting to the new living situation, until later in the year when Chandler decides to propose to Monica. He buys a ring, and just as he is about to start his speech, Richard walks in! (Damn big tree!) Seeing Monica reminds Richard of his love for her which he tells her the day after. Meanwhile, Chandler is getting scared that Monica suspects something, and he wants the proposal to be a surprise, so he acts disinterested until Monica starts to think that maybe Richard is the one for her.
Outraged, Chandler goes to Richard and declares that he was just acting uninterested to throw her off the trail, and that he was really planning to propose all along. Richard replies: 'Well, it worked really well'. Richard tells Chandler to 'go get her. And if you do, don't let her go.' Chandler scampers off to find Monica, when he meets Joey in the hallway. Joey tells him that Monica has gone off crying, and Chandler is devastated. He opens the door to find Monica surrounded by hundreds of candles waiting for him.
Monica tries to propose, but can't do it without crying. With some effort, Chandler proposes, and Monica accepts, and the crowd goes wild! (again)
Throughout the following year they plan for the wedding, until on May 15th, the day arrives. Now this is where I bow my head in shame because I missed the first half of this episode. I am so sorry!!! Well, I think what happens is that Chandler gets freaked out by the thought of being 'The Bings', and runs away. This isn't helped by the fact that Rachel and Phoebe think Monica is pregnant, which he overhears them discussing.
Eventually someone (I don't know who - sorry!) finds Chandler and convinces him to get over it. The marriage ceremony takes place, led by Reverend Tribbiani, and Chandler and Monica are now husband and wife. You may kiss the bride...

Chandler and Monica pictures