History of Half-Life Science

Aperture Science (which is the firm that developed the Portal gun) is in direct competition with Black Mesa (which is the firm for which Gordon Freeman worked) for military contracts

See the section Other facts on this link

Aperture Science and Black Mesa were bitter rivals. As seen in the Portal maps "testchmb_a_15" and "escape_02", Aperture Science employees were briefed on Black Mesa through slideshow presentations, such as one titled "Dollar$ and Sense: Competing with Black Mesa for DoD and Government-wide Acquisition Contracts"

EDIT: Here's an excellent guide on Half Life: http://members.shaw.ca/halflifestory/ Check out the timeline for an awesome detail of all events in the Half Life saga, and in the bottom a little speculation to its relation to Portal:

The date that the events of Portal take place aren't certain. We do know that GLaDOS says to Chell: "Are you trying to escape? [chuckle] Things have changed since the last time you left the building. What's going on out will make you wish you were back in here. I have an infinite capacity for knowledge, and even I'm not sure what is going on outside. All I know is I'm the only thing standing between us and them. Well, I was. Unless you have a plan for building some supercomputer parts in a big hurry, this place isn't going to be safe much longer." This implies that the game takes place during the Portal Storms or afterwards.

This site is not official but it's a great analysis IMO.

The must-play Half-Life games are Half-Life 1, then Half-Life 2. The expansions and spin-offs for Half-Life 1 which were not developed by Valve are quite skippable; Opposing Force, Blue Shift and Decay return to the setting and events of Half-Life 1, but portray the story through the eyes of a U.S. Marine, a security guard and two scientists (since Decay is a cooperative multiplayer game), respectively.

The Source version of Half-Life 1 is, and I quote:

a digitally remastered version of the critically acclaimed and best selling PC game

Needless to say it's smoke and mirrors, there's absolutely no significant difference between the two.

My advice is to play the original Half-Life 1, then Half-Life 2, and if you'd like to extend the pleasure, sequels Episode 1 and Episode 2 for Half-Life 2 are quite good, but not as good as the two main games.