How do I inflate this bicycle tire valve?
This is what's known as a Presta valve. It doesn't look broken, it looks like it is all there and the valve is in the open position because the captive nut at the very tip of it has been unscrewed. 

You can test if it is working by just nudging the end of it upwards briefly, some air should escape. This is good to do before inflating anyway to see that the valve has not become stuck before you pump.

You need a Presta compatible pump (some have a double head and some can be converted), or a small brass 'Presta valve adapter' to convert from a normal car valve (Schrader) type pump. 

A 'track pump' is a bicycle specific pump that typically has a head that suits both type of valve and is available from all good bike shops.

Here is an image showing operation of my preferred style of pump head, that I found online on multiple sites with no attribution (but probably Topeak themselves):