Logging amateur radio activity has been around about as long as amateur radio itself. At one time, it was required that all amateur stations kept a log of their activity. Now, it's no longer required, but it is still a very common practice.
Many people enjoy trying to contact a variety of different geographical areas, such as each of the United States, various different countries, or specific zones or grids. The easiest way to keep track of this is with a station log, particularly a modern online logbook.
There are a number of different online log books, but they generally have the same core features. They record your contacts, confirm your contacts, and track awards.
The first part is simple. You enter the station you contacted, the time of the contact, and the frequency or band. The system saves this info and you can review it from any web browser. Many logs will also let you record a little info about yourself, and show you the info added by other users.
The second part, confirmation, is done by cross referencing your log book with the other station's log book. So if I entered into my log that I contacted VE3XYY on 10 Jun, at 17:45, on 40m, the online system will then check the log of VE3XYY to see if he recorded the contact as well. If both stations recorded it, then it's considered "confirmed," commonly called a "QSL." It's important to note that in order for this confirmation to happen, both stations need to be using the same online log.
Lastly, many online log books also offer an award system. Common awards are "Worked all States," "Worked all Zones," and "DXCC." To earn an award, you must meet the criteria with confirmed contacts. To earn "Worked all States," for example, you must have a confirmed contact with an operator in each of the 50 United States. This is where choosing the right logbook becomes important; because only confirmed contacts count towards these awards, choosing a popular logbook increases the chances that the other station is using the same log, and thus allows for a confirmed contact.
The three most common online logs seem to be ARRL's Log Book of the World, QRZ.com, and eQSL.
Personally, eQSL is my least favorite. Personally, eQSL is my least favorite. I find the outdated interface to be confusing. While I enjoy the esthetic (see my webpage), it doesn't make for an easy logging experience. eQSL does go a little beyond just recording your logs, though, as the website also allows trading of "electronic QSL cards." This is fun, but it requires a subscription fee if you want to use a custom card or apply for awards.
The Logbook of the World (LOTW) is the ARRL's official online logbook. Rather than allowing you to enter contacts directly, it requires you to install a program on your computer, as well as submitting a copy of your FCC license. This is done to prevent fake accounts, but the hassle may also discourage some operators from using the service. I have heard that this system is particularly difficult for operators in other countries, so it may not be a good choice if you intend to make a lot of long distance contacts. LOTW also has an outdated-looking interface. Luckily, since you're required to use a program on your computer to submit contacts, you don't have to actually visit the webpage very often.
QRZ.com is my personal favorite. It has the added bonus of also accepting confirmations made by the Logbook of the World, so it greatly increases your chances of getting a confirmation on a contact. It also has the most modern looking interface of the three, allowing you add and review entries directly on the website, create and view profile pages, and even post messages in an online community. Here's my page. I also believe this is the only site of the three that doesn't charge a fee to apply for awards. It has a full API, as well, so you sync logs from a variety of desktop applications.
You don't have to choose only one, and there are many others out there as well, such a clublog, hamqth, hrdlog, and hamlog. Choosing a popular one is you best chance at receiving "confirmed" contacts, but if you're just logging for your own record, then simply choose the one you find most comfortable.
73!