the general belief is that lord ganesha is a brahamchari and not married. another story which is relevant to this argument is the story between tulsi ma and lord ganesha. as when tulsi ma proposed lord ganesha for marriage, then ganesha refused and said he won’t marry any woman. listening to this answer, tulsi maa got angry and she cursed ganesha that one day you will get married. so its quite possible that lord ganesha got married later on. but the image of him being a brahamchari remained more prominent in scriptures.
but in some parts of india, it is believed that lord ganesha is married to ‘buddhi’ and ‘siddhi‘. this story has a mention in one of the puranas. according to this legend, lord ganesha has two sons: kşema (prosperity) and lābha (profit). in north india the sons are often called as Śubha (auspiciouness) and Lābha (gain).
in the very popular hindi movie “jai santoshi maa” lord ganesha is shown to be married to ‘riddhi’ and ‘siddhi’ and having a daughter named santoshi ma, who is goddess of satisfaction. this story has no puranic basis but is generally accepted now.
in the very popular hindi tv serial “devo ke dev mahadev” also, lord ganesha is shown as married.
lord ganesha – photo from wikipedia by ahijitsathe
in the popular ashtavinayaka temple in morgaon (maharashtra), lord ganesha as ‘shri mayureshwara’ is shown with his consorts ‘buddhi’ and ‘siddhi’.
so there are many versions where lord ganesha is married to riddhi and siddhi or to buddhi and siddhi.
riddhi represents – prosperity and wealth
buddhi represents intelligence and wisdom
siddhi represents occult and spiritual powers. but here it will also represents samadhi or awakening of kundalini shakti as lord ganesha controls muladhara chakra where kundalini shakti resides.
story of lord ganesha marriage – when lord ganesha and lord subrahmanya reached marriageable age. then lord shiva and ma parvati thought of getting them married. when both of them came to know this, then they wanted to get married first and started arguing.
seeing their arguments, lord shiva told them whosoever will come back here after taking seven rounds of mother earth will be married first. listening this lord subrahmanya immediately set off on his vahana peacock with great speed to go around the earth quickly and return earlier than ganesha.
lord ganesha who is the deity of intelligence started worshipping lord shiva and ma parvati. after the worship, ganesha circumambulated seven times around lord shiva and ma parvati. ganesha also made obeisance to them seven times.
when lord shiva asked him why he is not going around the earth. so lord ganesha replied that by going around you i have taken 7 rounds of whole world. as you both are center of this world and control all the worlds. so i have completed the seven rounds first. so i should be married first.
on hearing the lord ganesha’s intelligent words, shiva and parvati ma were pleased. they decided to celebrate ganesha marriage with lord brahma beautiful daughters riddhi and siddhi.
editor’s note – the story represents that lord ganesha is the master of buddhi (intelligence), riddhi (prosperity) and siddhi (spiritual attainment). generally ma laxmi (deity of prosperity) and ma saraswati (deity of knowledge) does not stay together in one house. either you have knowledge or you have prosperity in the house. if you want both prosperity and wisdom together, then you are suppose to worship lord ganesha.
good links from web on this topic:
2. the marriage of lord ganesha (siddhivinayak.org)