The basic rule is that the verb ending –ize is an indicator of American English while the form –ise is British. However, it isn’t quite as simple as that.
Firstly, there are some verbs where the only correct ending is –ise. Some of the most important of these are: advertise; advise; apologise; devise; merchandise; improvise; revise. In contrast, capsize can only be written with the –ize ending.
In other cases, you’ll see –ize in American texts but both the –ize and –ise forms are generally considered acceptable in British English (e.g. recognise or recognize, criticise or criticize). British broadsheet newspapers seem to favour –ise, as though they’re trying to emphasise their British English credentials. As I’ve shown in the previous sentence, I also favour this, simply because it’s what I’ve grown up using, but plenty of good British writers do use –ize.
Strictly, -ize is more correct for words of Ancient Greek origin [-izein], though we British seem to have forgotten this now. Those words for which –ize isn’t an option tend to derive originally from French.