In T4 core rules (only, no Psionic Institutes):
Skill in psionics is on the same scale as other T4 skills, and earned/raised in the same way.
Your skill level is your "safe expendature" of psionic points on a single action. You can spend more, but you risk damage to your psionics stat if you do.
There is no roll to activate; spend the points, and the action happens.
Low point cost actions are short ranged and/or pretty weak; increased points get more profound effects (often exponentially) &/or longer range (separate points increases for range and power).
A skill 6+ Telekinetic, for example, is quite dangerous to a person.
A Skill 12+ telekinetic is dangerous to ships, or dangerous to people at artillery ranges.
T4 Psionics with Psionics Institutes:
As above, but add 10+ specials, and require a skill roll to activate abilities.
The Skill roll difficulty is based upon the quality of the Psionics institute one was trained at.
The Institute might also have non-standard chances for which abilities one gets to open.
A quick narrative description comparing editions
Zhodani Teleport Marines
In CT: They can get aboard, but usually can't get back, and might be able to bypass a door. The whole squad arrives together every time.
In MT: They all try at once; most of them arrive, and the rest try again next turn. They can usually skip a door, sometimes two. Sooner or later, the whole squad gets there. They are, however, all GM specials, since the CGen never came out for them.
In TNE: They all disappear at once.... and pop back in at various points along the line between the two ships. Eventually, they will all get aboard. They will bypass almost any solid obstacle (except cryogenic fluid compartments) until the Peep (clairavoyant) gets taken out. When done, the survivors then proceed back to their ship, again in a string of drops moving closer and closer. Along the way, the Teep will psionic assault anyone close enough after the peep shows them to him. THEN they go in.
In T4, no PI: They all arrive together. They can usually bypass a door, maybe two. Generally, they won't be able to get back without a dose of Special, and if they do, they risk damaging their Psionic ability. They tend to be less combat skilled than line troops, since they changed career from psionicist to Marine after a term or two.
In T4+PI: Most arrive together. Most will make it. The ones who arrived first will be able to bypass a door or two. They generally won't be able to get back, but a dose of special will again take care of that, at the same risk, elevated from the expenses on failed rolls.
In GT: The Jump marines are about 200 points to be able to make it ship-to-ship, and can generally bypass doors once aboard. The 100-point Marines are flown close by Navy cutters, and once aboard, can bypass doors. Eventually, tho', they tire out. The Teep and Peep usually are the same guy. Most will arrive together, but a few will arrive late, and a couple may pass out from repeated failures.
In T20: The Teep and Peep are separate guys, and are serious veterans. Most fo them are veterans, and they are not anywhere near as good at combat as equivalent experience (and thus Character Level) Marines, and are owhere near as good as straight psionicists...
In Hero Traveller: They count as having a "STOP" power, and are about PC-level. They can get over, bypass a few doors, and if they can take some time to rest (a few minutes, really), they can get back, too.
