People typically make partitions for home, usr, var and boot directories. Swap partitions are usually created by default. You can do without a swap partition and put everything into a one partition. Your distribution will complain and the system won't work well. The minimum number of partitions you can get away with is three, a root, a swap and a home. Anything fancier is done for performance reasons but not required.
The size of these partitions varries depending on the type of system you are building and what you will use it for. You want to have twice as much swap space as RAM. For the average desktop system with bells and whistles will need 3 GB for /usr, 500 MB for var, 300 MB for / and as much as you want for /home. You can see how much space you are using with the command "df -h". "mount" will tell you what file systems are mounted. Any Linux system will fit in five gigs of hard drive space, but they can be stripped down to less than 100 MB. Feather Linux fits into 128MB of live CD but has full Graphical environment with networking.
Here are some systems of mine: