I would like to write a scipt that takes a given date as as argument and prints to screen a number of events listed for that day.

For example I have a database of events (.txt file) that are of the form "12/12/12 :Jane's Birthday". I would like to write a script that takes the input "12/12/12" and returns the output "Jane's Birthday". I understand how to write a script that prints the entire database of dates and events but I do not know how to selectively print an event for a given date. Would anyone have any examples I could follow? Many thanks.

Try grep. For example,
will print all lines from file.txt that begin with the string "12/12/12"

I don't think you need to reinvent the wheel. You might want to check out the calendar application, which appears to do pretty much exactly what you describe. On debian it's provided in the bsdmainutils package.

Caveat: I've never actually used it myself.