Playing Netrunner online, be it in a chat room, on IRC, or whatever, is very different than playing face to face. Typing out everything you say and see would be very troublesome and make each turn last way too long. For that reason we've adopted some fairly standard abbreviations and shortcuts. It would be easiest to explain it by an example. Below is an example of part of a game played between evilone (playing corp) and liberatr (playing the runner):

[evilone] d
[evilone] a1: Accounts Receivable
[liberatr] What does that do again?
[evilone] Operation. Costs 5*, I gain 9*
[evilone] a2: AR
[evilone] a3: Edgerunner Temps
[evilone] ET1: ice HQ
[evilone] ET2: ice R&D
[evilone] ET3: install ANU in SDF1
[evilone] done 0/12/0 (a/b/c)

Here's the corp's first turn. He started off with his mandatory draw, indicated by the 'd'. For his first action (a1), he played an Accounts Receivable. For his second action, another Accounts Receivable, this time abbreviated. It's usually best to write out the full name of the card the first time, just to make your abbreviations clear. There may be many different abbreviations for each card, so writing it out the first time makes things clear. Questions in the middle of a turn can really slow things down. Note also that bits are represented by an asterisk. Third action, an Edgerunner Inc. Temps. Each Edgerunner action is indicated by ET1, ET2, and ET3. ET1 is used to install a piece of ice on Headquarters, ET2 to install ice on Research & Development, and ET3 to install an ANU (agenda, node, or upgrade; the runner doesn't know which it is until it is rezzed) on Subsidiary Data Fort 1. Each player ends their turn by indicating how many agenda points, bits, and cards they have (a/b/c). At this time, you would show common courtesy by reminding your opponent how many tags, virus counters, etc. you have.

[liberatr] a1: run HQ
[evilone] Ok. I rez Filter. Codegate, STR 0, @end run.
[liberatr] hmmm ... ok, I'm feeling lucky, a2: run R&D
[evilone] I rez Code Corpse :). Sentry, 5 STR, @1 brain damage, @1 brain damage, @end run.
[liberatr] Owie! Ok, 1-5?
[evilone] I'll take 2 & 5.
[liberatr] I lose my Shaka and a Nesgroup Filter.
[liberatr] a3: play Valupak Software Bundle
[liberatr] VSB1: install Skeleton Passkeys 3*
[liberatr] VSB2: install Viral Pipeline 3*
[liberatr] a4: run HQ
[evilone] 0* to pass Filter, you continue?
[liberatr] Heh. I pay the 0*, and yes, I continue. :)
[evilone] Okay, you're in. I have no cards.
[liberatr] Doh! That's right! But at least you have a Socket counter as well.
[liberatr] 0/0/0 (2 brain damage, you have 1 Socket Counter, SC, on HQ)

This should be pretty self-explanatory. The only new thing here is that the corp had to "draw" cards from the runner's hand.The runner asked 1-5?, and the corp selected from those. Making a run takes a bit more communication back and forth. For each piece of ice the corp says what it is (listing the stats the first time), then the runner indicates whether or not he can break, if he does, and how much he'll spend to do it.

There are a few more things, but I think this will get you going. Everybody has their own style of shorthand. It's just something that you'll have to work out for each game. The main thing to keep in mind is to be patient, and to ask questions if you have them, and to answer them if necessary.

I hope this helped. Good luck!