If this happens, then Person B will be able to spend person A's bitcoins. BUT there is only two ways this can happend:
a) Person B generate the same keypair (private key) as person A
or b) Person B generates a different keypair, which (public key) hashes to person A's address (a hash collision)
Take a look at the specification for at bitcoin address: https://en.bitcoin.it/wiki/Technical_background_of_Bitcoin_addresses
A Bitcoin address is a 160-bit hash of the public portion of a
public/private ECDSA keypair
For "a" to happen, person B would need to break "secp256k1".
I am no expert on this field, but from what I can find on bitcointalk.org, then the "strength" of this is 2^128 bits. And if you're wondering how strong that is, then look for the video "Exhaustive search attacks" from Dan Boneh. 11 minutes in the video he says "anything that's bigger than 2^90 is considered sufficiently secure" (its about attacking 3DES, which is a symmetric block cipher so I'm not sure it actually apply to secp256k1, but I really have no better argument for why 2^128 is "so secure that you won't never, ever, ever generate the same key as someone else"
And for "b" to happen, you would need to find a collision on RIPEMD-160(SHA-256(pk)). As far as I know, then neither RIPEMD-160 nor SHA-256 is known to have any vulnerabilities to hash collision attacks. So somehow breaking both or randomly generation a key which hashes is exactly the same as another bitcoin address, is highly unlikely.
There is also a chance for you computer to catch on fire, and some of the materials to melt together into a lotto coupon with winning numbers on (and a valid barcode), but it just won't happen because of the chance is so unbelievable small (it's the same with the "click and generate another persons bitcoin address").
EDIT Woops, one important thing: This all assumes that all bitcoin addresses is properly generated using "true random". All brain-addresses and addresses generated using a bad PRG, might be easy to find by either using a flaw in the PRG or exploit the (flaws in the) human brain.