Like the other answers have pointed out, your IP address is an identifier for your computer (or, more precisely, for your point of contact with the Internet). The best analogy, though, is probably a phone number.
A phone number (in the US) has up to four parts - an optional country code, a (used to be optional) area code, a prefix, and a line number. If your number is (123) 456-7890, then there are a bunch of numbers in the same neighborhood / service provider that all start with (123) 456. There are 10,000 possible numbers attached to that area code and prefix.
Similarly, an IP address breaks down into four parts - AAA.BBB.CCC.DDD. A block of IP addresses will be given to an internet service provider to hand out to users that connect to the internet through it. So, for example, your internet service provider might be 111.222.255.XXX. Each segment is a number between 1 and 255, and the 255 numbers that fill in the XXX make up a "subnet" (a network underneath 111.222.255), and your individual computer will get a random number assigned to it when you make a connection.
If you connect to the Internet through a wireless router, then the router itself is connecting to your ISP and it receives an IP address. If you use three different laptops to connect to that router, then all of those laptops will have the same IP as far as the outside world is concerned. However, you also have a special local network address (typically 192.168.1.XXX) that would let the computers on your home network talk to each other.
Your ISP has records of who is assigned a specific address. So they could use the IP address to trace an electronic communication back to you specifically. Other people may be able to see your IP address, but at best they can use that to determine what geographic region you're located in. They would need to contact your ISP (and presumably get a warrant) to find out who, specifically, you are.