How do viruses spread?

When you execute program code that's infected by a virus, the virus code 
will also run and try to infect other programs, either on the same computer 
or on other computers connected to it over a network . And the newly 
infected programs will try to infect yet more programs. 

When you share a copy of an infected file with other computer users, 
running the file may also infect their computers; and files from those 
computers may spread the infection to yet more computers. 

If your computer is infected with a boot sector virus, the virus tries to 
write copies of itself to the system areas of floppy disks and hard disks. 
Then the infected floppy disks may infect other computers that boot from 
them, and the virus copy on the hard disk will try to infect still more 
floppies. 

Some viruses, known as 'multipartite' viruses, can spread both by infecting
files and by infecting the boot areas of floppy disks.

Article from COMPUTER VIRUS FAQ