Since a byte is 8 bits, an integer data type can hold 2(8*bytes).
Let's run the numbers:
A tinyint can hold a value as large as 28. 28 is 256, and a tinyint doesn't use negative numbers,so it can hold values from 0 to 255.
A smallint is 2 bytes, or 16 bits. 216=64K, so a smallint can range from a minimum of -32K to a maximum of 32K-1. The calculation of the exact values is left as an exercise for the reader.
An int is 4 bytes, so we're talking about 232. I know this one by remembering a 32-bit CPU can address 4GB of RAM, which makes the range not 0-4G, but -2G to 2G-1.
Finally, a bigint is 8 bytes. 232 is...a really big range. According to Books Online, the minimum value is -9,223,372,036,854,775,807. The bigint tops out at 9,223,372,036,854,775,808.
As always, refer to Books Online for more detail.