What exactly are tax allowances?
The number of allowances you put on your W-2 determines how much tax is withheld from your paycheck. The more allowances you claim, the less tax is withheld from your paycheck. Essentially the payroll software will use either tables published by the IRS or internal calculations to estimate how much tax you will owe for the year based on the number of allowances you claim and your pay rate, and withhold enough tax to cover the pro-rated tax amount for that pay period.
It is not tax paid exactly - you pay your tax when you file each year (or quarterly if necessary). When you file, your income and deductions for the whole year are used to compute how much tax you owe for the entire year. The amount that is withheld is then subtracted from that amount to see either how much you still owe or how big a refund you get.
The reason it's important to get allowances right is so you don't under-withhold and owe too much tax when you file, which might cause you to incur a penalty.
How many should I claim, if any at all?
You should almost certainly claim at least some allowances, since you get a personal tax deduction for yourself. If you are married and/or have kids or other dependents, then you can claim even more. If you answer the questions on the form accurately, and you don't have any unusual tax circumstances (untaxed income, etc.) that would cause your tax to be based on much more than your actual income, you should be fine.
If you are concerned, the way to determine if you are withholding correctly is to take how much is being withheld, extrapolate that out to the end of the year, take your income and estimated deductions, figure out roughly how much tax you will owe (regardless of withholdings) and see if enough is being withheld to cover either 90% of the tax due or 100% of tax paid last year, or ensure you will owe less than $1,000 in tax (those are the limits for underpaid tax). If that sounds very complicated, it can be, but for most people a roughly accurate W-4 (or equivalent state form) is enough to make sure taxes are withheld properly.