There are a lot of events in Cambridge that require a certain type of dress, the following is a fairly complete list:
Smart Casual:
This comes in many forms, but a minimum standard is:
Him: Long trousers (not jeans, cargo pants or other casual wear); short or long sleeved shirt with collar; closed/hard shoes (not running shoes or sandals).
Her: Skirt or trousers with a nice top and nice shoes or boots.
For some college events, such as the Matriculation Photograph and Formal Hall, you will be required to wear an academic gown. These can be bought at Ryder & Amies or Ede and Ravenscroft, either new (~£100) or ex-hire (~£50). The Graduate Union also rents gowns on a daily or weekly basis. Gowns are also available at discount from the MCR at the beginning of every year (~£30), although these are limited!
Lounge Suit / Cocktail Dress (‘Smart’):
Him: A normal suit, with necktie.
Her: A dress ranging in length from just above the knee to about two inches above the ankle.
Black Tie (‘Semi-formal’):
Him: A dinner jacket (‘DJ’, or tuxedo in American English), and a black bow-tie. Dinner jackets are black and usually have satin lapels. The accompanying trousers have a single satin stripe down the outside. Shirts may be either wing tip or full collar. A number of clothiers in town hire out dinner suits at around £25 for the jacket and trousers. Black socks and shoes. Waistcoats if you can.
Her: An evening gown or cocktail dress.
White Tie (‘Formal’):
Him: A black tailcoat, a white waistcoat, a white stiff-fronted shirt with a stiff winged collar, and a white bow-tie. Only a few events require white tie, so you’re not likely to need it.
Her: A ball gown that reaches at least to the ankles.