The thing to the right (from phulbari.com) is called a handbag in BrE, and a purse in AmE. One can say handbag in AmE, but it sounds rather old-fashioned. In keeping with that feeling, I'd tend to reserve the term for vintage items (when speaking in American environs). Handbag can be used to refer to most handled women's accessories for carrying around life's essentials—money, lipstick (lippy: BrE informal, orig. AusE), (BrE) mobile/ (AmE) cell phone, Syndol--which itself is a major reason to emigrate to Britain. Longer-handled ones might also be called shoulder bags (as they could be in AmE as well). But in everyday BrE life they all tend to be called just bags--as in I have some Syndol in my bag--want some?
Purse in BrE is a (typically women's) leather/cloth/etc. thing that money goes directly into--like the ones at the left, from Arnold & Arnold. Thus female BrE speakers usually have purses inside their bags. AmE retains this sense of purse in change purse. For North Americans, the things on the left are wallets. If it's in a man's pocket, it's wallet in both dialects--but my dad (like others in his AmE-speaking generation) calls his a billfold.
BrE has a few handbag idioms worth noting. Handbags at dawn (also a great name for a band) or handbags at ten paces is a way of referring to a usually loud, public fight--originally among footballers. This is sometimes shortened to handbags. The OED's earliest citation for this is 1987, but they're looking for earlier ones. To handbag is an established verb in BrE, meaning 'to assault with a handbag', and can also be used figuratively, meaning 'to verbally assault or criticise', as in:
Not since Mrs Thatcher handbagged her cabinet into attending a seminar on climate change at Number 10 had so many senior Tories been seen doing something green in one place. -- The Telegraph
Kate's Canadianness has reminded me that I haven't reported an instance of being assumed Canadian. It happened last weekend at the Scrabble tournament, though to be fair it was after I was explaining the differences between Canadian, American and British spelling. Who but a Canadian would know such things? I would, apparently.
(Links to commercial sites here are just (a) to acknowledge the sources of photos and (b) prevent people asking me "where did you get that bag/purse?" Now you know already. This is not an endorsement of these companies/products, but they are rather pretty, aren't they?)