Your connection speed to the Internet will depend on what type connection you have and what speed you are paying for with it. The fastest speeds are available using cable connections with DSL the next fastest, satellite the next (but will vary with weather conditions) and the slowest is dial-up. Most people are using cable or DSL now. Speeds are measured in megabits per second but usually they are just said as so many meg, and dialup is in kilobits per second. You can get slower connections of about 1.5 meg for a lower price. Higher speeds on DSL usually run about 6 meg. I have seen cable using fiber optic lines run speeds for residential up to 20 meg and they say they can sometimes supply for a price up to 100 meg. Cable using copper wire usually tops at 6-10 meg
Different things can affect your speed of your connection. Depending on how the system was designed, the time of day (how many people are using), and the type of media can affect it. The quality of the wiring will also affect it.
There are a number of sites on the Internet that will measure the speed of your connection. The two I like best are speakeasy.net and speedtest.net. Both have relative easy interfaces to understand. However be careful that you are choosing to run the test and not to scan your computer for other stuff or to download some program. At speedtest.net click the Begin Test button in the laptop on the screen. At speakeasy.net you will choose a city and that starts the test (I usually use Atlanta). Then you will get the results. You will get an upload speed and a download speed. When most people talk about connection speeds they are discussing download as most people have the most information sent to them and they send very little to other locations. Gamers and people that upload lots of pictures are the big exceptions.
The upload speed will usually be a lot smaller than the download speed. It is possible to get connections that have the same upload and download speeds, but they are expensive.