|The arms of one branch|
of the Cromwell family.
|The arms of a branch of|
the Tattershall family
The first answer is the ever helpful 'it depends', but like most bits of heraldry, it is largely driven by some core rules. The various forms of combining arms are known as marshalling.
To be more specific (and note, these rules are most common in British heraldry): if two people whose families are both entitled to bear ams, marry, their respective coats of arms are impaled, which is to say placed in the left and right (dexter and sinister) halves of a shield divided per pale, with the male arms on the dexter.
female Tattershall. Cromwell is argent, a chief gules, over all a bend azure. Tattershall is chequy or and gules, a chief ermine. Note to the observant reader - yes, we didn't mention chequy - that comes later, but you can guess. While they are both alive, they'bear each others' arms impaled: per pale: argent, a chief gules, over all a bend azure; chequy or and gules, a chief ermine. As a shorthand, it could be referred to as per pale, Cromwell and Tattershall.
The husband would have used the arms on a shield (or escutcheon), If the lady had to have her arms depicted, they'd be on a lozenge, i.e. a diamond standing on one point. (This applies generally, not just to impaled arms.)
Formerly, instead of being impaled, arms were sometimes dimidated, in which the dexter half of one was placed next to the sinister half of the other. The end result could be oddities like a lion with a fish's tail, which may have been one of the reasons it was dropped.
There's a special case of this - if the lady was due the arms in her own right, i.e. was a heraldic heiress, for example because she had no surviving brothers, her husband would generally not display his arms and hers impaled. Instead he would display hers on a shield on top of his to indicate that he and his descendants would claim those arms, known as an escutcheon of pretence. Annoyingly, the free version of Coat of Arms Studio won't let me draw this for you, but you can work it out.
Now, their children won't display those arms. Instead, they'll display the father's, under typical English rules, except if their mother was a heraldic heiress, in which case they could display their parents' arms quartered, with the father's top left/bottom right. Quarterings can get considerably more complex as time and inheritance goes by, of course, and often lesser arms get dropped or edited out. See the Wikipedia entry on marshalling for more detail and a couple of extreme examples.
If you want a brilliant example and practical tutorial on this, go take a trip to Tattershall Castle in Lincolnshire, go up to the top and walk down through the various levels of the keep, looking at the windows. From top to bottom there's a sequence of arms covering multiple families, marriages and inheritances with all manner of quarterings, impalements etc. Fantastic fun.
 They'll display something different while their father's alive, but we'll cover that next time!