To use the internet you have to be connected to it, obviously. For most people this means paying for an account at an internet service provider (ISP), which gives you access to the internet through some means (telephone dial-up, telephone line ADSL, data over a cable TV system, wireless microwave or satellite links, etc.), and they have to make their computer connect up when they need access (the computer doesn't magically know when it needs to connect, even if it can automatically start connecting in some circumstances).
Some connection methods, such as dial-up, involve using something (the phone line) to the exclusion of something else (your telephone), or charge you fees for the duration that you're using them, so you only connect to the internet as required, disconnecting when finished. Others methods can work simultaneously with other equipment on the same line (e.g. ADSL or “cable” systems), have their own dedicated connection, or don't charge time-based fees, so you can leave them permanently connected.
It's important to know what sort of account you have with your ISP. If yours charges for the time you connect to them, then ensure that you disconnect when finished. If yours charges you for the amount of data that you use, then limit your activities on the internet (downloads, web page browsing, etc.). Staying connected longer than necessary, or excessive uploading and downloading, can be very expensive.
All data that comes down the connection towards you is a “download”—this includes e-mail that you receive, webpages you look at, etc., not just files that you download and save. And data going in the other direction (from you) is an “upload”—this includes e-mails that you send, your requests for webpages, and files that you send.
Most ISPs that charge for the data going through your connection will include all data traffic, though some don't charge for traffic that comes from some of their own services (e.g. they might run their own private game servers, or file repositories), and some don't charge for uploads.
Even if you have a truly unlimited type of service, staying permanently connected means that your computer is exposed to the internet more than it needs to be. So if you're not sure that you're adequately protected against malicious people on the internet (of which there are huge numbers) you're best to disconnect when you don't really need to be connected.
Most ISPs provide more than just “access” to the internet, they may also provide you with an e-mail account, personal webspace, and other features.