How can I get stains out of a pine table?
Some background, there are two types of wood water stains. White rings and black marks. White rings form when the water has only made it into the finish, this is the "best" kind of stain to have. Black marks occur when the water has penetrated into the wood itself. This is slightly harder to fix and will ultimately require either sanding or covering it with a richer, darker stain, although sanding is the only sure-fire method.
There are some options that involve staining or not, I'll start with the least invasive. 

White ring removal


Place a dry, clean white lint free shirt or bed sheet over the water stain and rub with an iron at medium setting for a few seconds, then lift the cloth and repeat until the stain is gone. You want to be rather brief in the application of the iron as not to further damage the table. You can use a hair drier too. This works very well on antiques with delicate finishes that you can't replace.
Apply lemon oil to fine steel wool, make sure it's fine, and gently buff the rings out.


Black mark removal


Stain the wood with a suitable, darker stain. This is a judgement call, if the black mark is very pronounced or covers a large area you are better off sanding. You can test this out because if this doesn't work, you are sanding anyway.
Sand the table down to the wood and begin bleaching. Use a brush and apply bleach to the black marks as best you can, let it sit for a few hours and reapply until the stain is gone. If normal bleach doesn't work get some commercial wood bleach. Follow the instructions but you should have two bottles as part of a kit. After application it should start bubbling. Before finishing and project and restaining, you MUST apply a mixture of 2 parts water and 1 part vinegar to the area you bleached to neutralize the bleach, then rinse with normal H20. This is to ensure a good surface to re-stain.