Let's see:
Jinteki: There is an archetype called Cambridge which makes a lot of exposed servers, many of which are traps, aiming to either kill a runner who is not cautious or (less often) out-bluff a cautious runner and score out on agendas that have been sitting open. It has a lot of minor variations; here is one example from netrunnerdb: http://netrunnerdb.com/en/decklist/24359/more-generic-cambridge-top-16-nationals-2015
NBN: There are basically two main archetypes. One is killing, the other one is fast advance.
The kill decks use NBN's strong native ability to tag the runner and import Weyland tools to actually kill them. The currently popular version of this is called Butchershop. A very simplistic version can be made with one Core Set (preferably two, so you have 3 Scorched Earth and 3 AstroScript Pilot Program). A vast improvement can be made by adding Opening Moves (for Jackson Howard) and Future Proof (for Project Beale).
The fast advance decks are pretty straightforward: if you have an AstroScript token, then you can score a 3 advancement agenda, usually either another AstroScript or Project Beale, from hand. You can then use SanSan City Grid (from NBN) or Biotic Labor (from HB) to help score other agendas (for instance the first AstroScript) from hand. Alternately you can just use an actual remote.
HB: HB varies a fair bit, but it's more frequently fast advance or glacier rather than kill. The fast advance style is basically to score Project Vitruvius (from Cyber Exodus) and/or Accelerated Beta Test from hand, using Biotic Labor.
There are lots of spins on glacier, with not so many key cards to make it work as in most other archetypes. A very thematic way to build HB glacier would be with a ton of bioroid ice, perhaps using the ID Stronger Together (from What Lies Ahead). I wouldn't recommend using all bioroids, simply because it is too hard to leave an agenda safely in a remote that way.
Weyland: Until Order and Chaos, Weyland was mostly kill and glacier. Weyland kill is just the reverse of NBN kill: Weyland's tools do the killing, NBN's tools do the tagging. Weyland glacier is primarily based on big barriers, some of which are advanceable, and Archer.
With Order and Chaos, Weyland got the ID Titan Transnational, which makes it possible to run an "Atlas train" (i.e. scoring 3 Project Atlas from What Lies Ahead in quick succession). This basically created the Weyland fast advance archetype.
Anarch: Noise is an archetype unto himself. There is a related weird archetype using Valencia (from Order and Chaos). The other Anarchs often uses fixed strength breakers with Datasucker and/or Parasite support, and tend to get economy from out of faction.
Criminal: The Core Set and Honor and Profit hint at an "intel" Criminal deck, which is constantly poking around and getting information without making runs, and then actually making runs when they know they can snipe an agenda. The original card supporting this archetype is Lemuria Codecracker. This archetype was apparently never very good, but it is thematic and kinda fun.
More commonly, Criminals tend to go with a tactic where they are constantly running, using various cards, especially Datasucker, to gain benefit from doing so. This was very popular out of Andromeda (from Humanity's Shadow) for a long time, since she has an extremely powerful early game.
Shaper: Shapers are traditionally big rig. It is pretty straightforward to see how big rig works when it works, so I don't think this needs so much comment. Kit (from Creation and Control) mixes things up a fair bit in that she has a strong early game, needing only a Decoder to get into a server with just one ice.