My first time travelling and I have recently booked a flight with American Airlines. On my ticket it says "Baggage Allowance:1PC." What exactly does this mean?
Your booking should give the precise detail. Normally airlines will allow some amount of checked baggage (looks like in your case that's 1 piece) and have some additional hand-luggage allowance, often one smaller bag and 1 personal item (eg. handbag or laptop).
However the precise rules vary from airline to airline, and also may depend upon the exact ticket you booked. Some airlines have really cheap tickets that have no checked baggage allowance. European airlines and American airlines have slightly different rules for dimensions and weight of hand-baggage. Airlines also vary in how strictly they enforce these rules - some budget airlines are very rigorous in the checks they perform. In many years of flying with many airlines I have never had my hand-baggage weighed.
In all probability with American you have 1 piece of checked baggage, one small suitcase and one personal item as hand-baggage. However you should look carefully at the ticket or itinerary for your booking and understand the exact rules for this booking. All the information will be available though it may take a few minutes of following links to find it all.
If your checked baggage is likely to be heavy I strongly suggest that you get a little scale that lets you easily check its weight. Checked baggage does get weighed at checkin and breaching weight limits will incur charges.
"PC" is an abbreviation of "piece". It means you can check-in one suitcase (or similar).
Checked-in luggage is luggage you hand in at the airport and then goes in the luggage compartments (I suddenly can't remember what they are usually called) of the plane, and then you get it back after arrival. - There is no way to get to items in it while the plane is in the air. There are some limits to the size of checked-in items, but almost all regular suitcases will work. Airlines have different size limits so you'll need to check their documentation (or consult someone with experience with American Airlines, which I don't have) for details.
In addition you're probably allowed to have some carry-on/hand luggage, the size limits are usually a lot smaller than for checked-in luggage. Again airlines have different rules, so check what AA allows. Some/many airlines (and I don't know if AA is included) allow two pieces, where one should be a laptop bag (again there'll be size limits - they probably don't check there's a laptop in there, but the rules I remember seeing specifically said "laptop bag).
The baggage allowance refers to luggage you can check at no additional charge with your ticket. You can purchase additional allowance if you need to. 1 PC means that you are allowed one piece of luggage.
Your checked bag needs to weigh 50lbs or less and measure at most 62" linearly which means the width, depth and height added together cannot exceed 62". Here is the official page describing this policy.
Additionally, you can take a carry-on with you plus a personal item. The personal item is usually a purse, camera bag, laptop bag or briefcase but it practice it can be anything which falls within its own limit of 18"x 14" x 8". Limits for carry-on and personal items are described here.
Also worth nothing is that certain items are not allowed in carry-on and others are not allowed in the checked luggage. Restrictions are described in both pages linked above.