What is the V-shaped thing called one puts in a checkbox?

You might be looking for "check" or "tick", or "check mark" or "tick mark".

Do you mean a check mark (also called a tick)? http://en.wikipedia.org/wiki/Check_mark The x-shaped mark can be called simply an x or a cross.http://en.wikipedia.org/wiki/X_mark

In the UK we were usually told to put a tick or a cross in the appropriate box...

...but if you follow the link and switch between British and American corpuses, you'll see that the "relative prevalence" (number of instances per 100M words) for BrE is about 8 times higher than that for AmE.

Just a guess, but I wonder if maybe the (historically) higher proportion of immigrants less familiar with English (for reading the instructions on filling out the form) might mean that US forms were always more likely to expect just a single mark (tick or cross) to mean "this is the one that applies".

Some years ago a friend who helped out counting the ballot papers after a General Election told me she was amazed at how many were classed as "spoiled" because the voter had put a tick against their preferred candidate, and a cross against each of the others. The intended vote was obvious, but they weren't allowed to count it. We both thought this probably caused disproportionate (hopefully, unintended) disenfranchisement of recent immigrants who might not understand the instructions.