This is a basic guide to plexing. There's a bit more to system occupancy tactics than this, and of course this guide only covers the mechanics of how to capture a plex and does not deal with plex fighting, which is an entirely different kettle of fish. However, it will at least explain the basics.
If you are not familiar with the basics of the plexing system, it is at its core very simple:
1. Plexes must be scanned down with your onboard scanner. You do not need probes, and if there is a plex within the 64AU range of your scanner you will immediately receive a 100%, warpable hit. When you warp to your scan result, the plex will open and appear on the overview. In most systems, the onboard scanner range is large enough that you will only need to scan once to cover the whole system. In some very large systems (Halmah, for example) you will need to move and do it again to make sure you've caught everything.
2. There are offensive and defensive plexes. Defensive plexes (for us, named things like 'Minmatar Outpost') contain only friendly rats. Don't shoot them; they have the same red cross appearance on overview as other rats but they won't shoot you unless you start it. (Then they get annoyed and shoot back.) Offensive plexes (Amarr Outpost, Stronghold, Facility etc) contain hostile rats.
3. Plexes come in three sizes: minor, medium and major. There is a further division in that major plexes have two subtypes, namely restricted and unrestricted. You can tell what size plex it is by its name: minor and major plexes say so in their names, and thus anything that doesn't have a size indicator in the name is automatically medium. Once a major plex is open, it will say in a column on the overview if it's restricted or unrestricted. The size of the plex determines which shiptypes are allowed inside, as follows:
Minor: T1 frigates, T1 destroyers. Faction frigates (both empire and pirate) count as T1.
Medium: T2 frigates, T1 cruisers, as well as the smaller stuff named above for minors. Faction cruisers count as T1.
Major (restricted): T2 cruisers, T3 cruisers, T1 battlecruisers, as well as smaller stuff named above.
Major (unrestricted): Anything.
All plex types except Major Unrestricted have accelleration gates. It is sometimes possible to get stuck on the gate after warping to it. To avoid this, warp to the gate at 10. You will land within activation range of the gate (due to the fact that its 'you're on 0' radius is strangely huge), but far enough away not to get stuck.
4. You capture the plex by orbiting the button (visible on overview, see attached image for clarification. If viewing with in-game browser and image appears cropped, maximise the window) within the distance indicated on the overview until the timer counts down to 0. Only people within capture range of the button when the plex closes will receive LP (in offensive plexes only. At present, you do not get LP for defensive plexes, although this may change in the near future). Once the timer reaches 0, the plex will close. It will not be able to despawn until you leave grid, however. Plexes of the 'outpost' type will respawn in the same size half an hour after the previous, closed one despawns.
5. In offensive plexes, waves of hostile rats will spawn. You may choose to kill these, or you can speedtank them. Amarr rats fire lasers (for obvious reasons). They are weak aginst EM damage.
I hope you find this guide useful. There's a few other tricks and nuances to plexing, but they are outside the scope of a basic guide such as this. Get yourself a cheap frig or destroyer (or ten!) and go out and conquer systems! If you have any questions, or would like to get involved with the plexing effort but don't know where to start, please get in touch. I, and the rest of FEO, would be happy to point you in the right direction.