Marriage
A Sim can propose to another depending on their relationship. If the other Sim accepts they will get married.
The Sims
In The Sims, anyone can get married. Either Sim can propose, but the proposer must be at home to do it. Whether the other Sim accepts or not depends on their mood or the relationship between the two Sims. If the other Sim accepts, however, both Sims will change into their formal clothes and share a dance together, and the proposing Sim will be charged §1000. The Sim who was proposed to will take on the family name of the other Sim and move in with them. Since there are no defined family ties or family relationships in The Sims, the option to get married is available to Sims who already live together as long as there is not already a relationship between them (example: two Sims are created in "Create a Sim" mode, but when its time to define their relationship, none is provided, so they have the same last name, but are neither related nor married to one another).
Same-sex couples (or two friends of the same sex) can ask one another to 'Move In' if they do not already live together. Whether the other Sim accepts depends on their mood, their relationship, and the quality of the other Sim's house. The Sim who moves in will take on the family name of the Sim who proposed, even if the two are simply good friends. (This is unavoidable, as Sims in The Sims do not, properly speaking, have surnames.)
Sims cannot get a divorce, but if their relationship is low enough, they can choose the Attack option and fight each other. Which Sims wins depends on their Body skill. The Sim who loses may choose to move out, and if they do, they will never be seen in the game again.
The Sims 2
When two Sims are in love they can get first engaged and then marry (with or without a wedding party). If the Sim to be married is not living with his or her partner yet they will move in as soon as the wedding ceremony is over. Either Sim can propose.
There are two ways in which Sims can get married. First off, both have to be engaged, and can either get married by clicking on the Sim in which they are engaged to and selecting the "Get Married" interaction, in which the two Sims exchange vows and kiss to signify their marriage. It also possible for two Sims to get married under a wedding arch without throwing a wedding party.
The other way, and more "traditional" way to get married is to have a wedding party. In order to do this, either Sim can click on a telephone, and select "Throw Wedding Party". This is basically like a normal party, only your Sims are expected to get married. You may or may not use the wedding arch. But a difference between normal parties and wedding parties is that any guest you invite will show up even if you have a low relationship with that Sim!
Like normal parties, a wedding party is rated by how good it is. So, it is possible to have a horrible wedding party. In order to prevent this, you can pretty much set up the wedding party like you would a normal party (having music, buffet table, bar, etc.)
Finally, when the marriage takes place, both Sims switch into wedding attire (which is their "formal" wear, so make sure to choose something wedding appropriate beforehand) and head beneath the wedding arch. Your guests gather around, and the Sims exchange vows and kiss to signify their marriage.
If the "Wedding Party" is successful a limo will come for the couple to take them on their honeymoon. While they are on their honeymoon, the newly-wedded Sims' need bars will have a significant increase.
The Sim who initiates the 'Get Married' interaction is the Sim whose last name both parties to the marriage will take. However, you can use a program such as SimPE to edit this, if you so choose.
Joined Union
Marriages between two Sims of the same sex are referred to as "Joined Union" in The Sims 2, although a Wedding Party is still called a wedding party. A Joined Union is the alternative to a marriage available for same-sex couples.
Like a regular wedding, when two Sims of the same gender are in love they can get first engaged and then enter a union (with or without a wedding party). If the Sim to be joined is not living with his or her partner yet they will move in as soon as the wedding ceremony is over. Again, the Sim who initiates the interaction is the one whose last name both parties to the union will take, though you can use a program such as SimPE to edit this, if you so choose.
Sim Unions provide fewer Aspiration points than marriage. However, a hack available at simbiology.com allows same sex partners to enter marriages with the same titles, wants and fears as with opposite sex partners.
The Sims 3
Sims who are in love can propose marriage and become engaged. They can then either have a "Private Wedding" or a wedding party.
When they wed the last name taken by the couple will be that of the Sim who did the proposing. If the Sim whose name changed has children, they will also take on the last name of their step-parent.
If you wish for your couple to get married, you have to build up their relationship until they are in love, and have a romantic interest. Then after they have a full relationship bar, you will get the action: Propose going steady, under the romantic section. Your Sims will then become boyfriend and girlfriend. When you then have a full relationship bar, you can propose marriage. Unlike in The Sims 2, where same-sex partners were only allowed to have civil unions, in The Sims 3 they can have a marriage the same as opposite sex partners.
It is also possible for a Sim to get married to a ghost in The Sims 3.
Divorce
Contrary to popular belief, it is possible for marriages in the Sims' world to end in ways other than death.
If two married Sims have their relationship with each other fall enough, such as through witnessing an affair one Sim can kick the other to the curb. To have the couple get a divorce, choose whichever Sim you want to do the breaking up, click the other spouse, and click "Break Up". This option will only appear if the active Sim's relationship is low enough with their spouse. The divorcee will then flee the lot and reappear in the Family bin in the neighborhood screen. They have no chance to take any children they have with them or remain in the house. If you want them to stay in the house living with their family after the divorce, you will have to move them back in.
Avoiding Divorce
If you want your Sims to avoid divorce you can either wait for them to stop being mad at each other or with The Sims 2 Apartment Life you can have them do a interaction called: Repair Relationship with... which will have the two Sims try and deal with their problems.