I am new to Netrunner, so if this question is stupid please forgive me. My question is this, as the Corp. if I discard a trap to my Archives and the runner accesses it, does it fire or does it do nothing?

Depends on what the card says.

Ex: Snare is no. Breached Dome is yes.

Not a dumb question at all, Netrunner has a lot of non-intuitive rules.

Basically, cards which trigger when accessed always work unless otherwise stated (such as Snare! like nungunz mentioned), or have their effect preceded by another card effect, causing the trigger condition to become invalid before it can resolve (such as Film Critic and Explod-a-palooza).