The couple first met backstage, three weeks after the death of her father Bruce Paltrow, at the A Rush of Blood to the Head tour in October 2002. He allegedly wrote the song 'Fix You' to help her deal with the grief when they first started dating.
The pair got married on the 5th December 2003, in a hotel in Southern California in a quiet ceremony that involved none of their friends or family.
They later had two children; Apple Blythe Alison Martin, born in 2004 and Moses Bruce Anthony Martin born in 2006. Paltrow explained Apple's unique first name on Oprah, saying: "It sounded so sweet and it conjured such a lovely picture for me - you know, apples are so sweet and they're wholesome and it's biblical - and I just thought it sounded so lovely and ... clean! And I just thought, 'Perfect!'" (Wikipedia)
Moses is named after as song that Martin wrote for their wedding day, which appears on Coldplay's Live 2003 album.
They announced in March 2014 that they were separating with a process they called 'conscious uncoupling'. In April 2015, Paltrow filed for divorce from Martin. It was all finalised in July 2016.
Source: Wikipedia and IMDB.