Correct, out of the box it'll work on AT&T, Sprint, or T-Mobile in the US or EE in the UK (I'm not sure if a US-sold model will work in the UK out of the box and vice versa, though - I believe they will).
For any other supported network in the world, you will need a SIM card from that network provider.
Suggestion - if you're in the US, T-Mobile is the best option if you have T-Mobile where you live and she plays, otherwise AT&T is.
Verizon's plans are ridiculously overpriced, based mostly on Verizon's over-inflated reputation (it isn't really true... Verizon is bloody slow from what I've seen) and Sprint is far more expensive than T-Mobile (though cheaper than AT&T), again for no apparent reason (Sprint isn't really better than T-Mobile... maybe a bit better coverage, but if coverage matters you'd probably get AT&T, and it's way slower than T-Mobile). In fact, at the most useful $30-ish level, Sprint is $5 MORE expensive than AT&T for 3GB - it's insane.