Underage, is, quite simply, anyone who's not old enough to do something.
The exact ages for different activities vary from place to place (state to state in the US), and are often riddled with loopholes.
For example, in the UK, anyone of any age may drink wine or spirits with a meal, but you must be over 18 to drink lager or bitter. (I'm not entirely sure what the law on alcopops is... what would they be classed as?). It's illegal to buy alcohol or for another to buy alcohol for you if you are underage, but it's quite alright to down that litre bottle of Bacardi that your older sister's been keeping in reserve. (Naturally, you wouldn't be quiet alright afterwards, but that's outside the scope of this node). In the US, it's governed by state laws, with some setting the limit at 18, others at 21.
Sex is quite complicated. In the UK, the laws are simple: shagging anyone under the age of 16 is statuatory rape - the child is considered to not be old enough to give consent. (If it's a pair of randy fifteen year olds, then the law doesn't explain what should happen.) In the US, the age of consent is also 16, but an adult may not have sex with a minor, who is anyone under the age of 18. This could cause some major inconveniences, so it is legal to have sex with a minor, so long as there has already been an established relationship whilst the adult was under eighteen.
In the UK, you have to be 16 to join the armed forces. It's sometimes quoted that You can die for your country, but you can't have a drink. This rule doesn't apply to the Merchant Navy, so while underage Prisoner of Wars were returned to Blighty if they were with the RAF, Army or regular Navy, the captors were quite entitled to keep 15 year olds in prisoner of war camps if they'd joined the Merchant Navy.