Psychic Mastery Levels -
Mastery levels are the new way to identify how many powers you psyker can use and what type of spells they can use. For every power they can cast in a given turn, they are that level mastery. I.e. if you can cast one power in a phase, you'r Level 1; if you can cast two powers you're Level 2. This is also how many warp charges you generate - each power unless otherwise noted costs one warp charge.
Psychic Defenses -
Defenses are a lot more important as psychic powers are likely to be a lot more common-place. Even in 5th they were important and not having them could prove frustrating. In 6th every army has some sort of passive psychic defense - i.e. if something targets them they can stop it. There are now very few aggressive psychic defenses which can stop self-targeting enemy powers.
So, every unit now has the chance to stop a psychic power on a 6+ called Deny the Witch. No matter what you do, you have this option and it's not reliable at all. Do NOT consider this sufficient. Pskers have been changed to improve this roll. Any unit containing a model with a psyker can add +1 to their Deny the Witch roll. This is improved to a +2 if said model has a better Mastery level than the psyker casting the spell. So on average a psychic defense based model is going to give squads a 5+ Deny the Witch roll - still not great but it gives psychic models some utility and it's roughly close to what psychic hoods were before in terms of stopping powers aimed at your units. Speaking of psychic hoods, they have changed to provide this benefit to any unit within 6" - again, roughly the same as before in terms of stopping power, just the range is much smaller and an aura and it's no longer an aggressive defense.
There are other types of psychic defenses which are aggressive though and two of them can be allied into most armies. These include the Runic Weapons of Space Wolves (a straight 4+ against any power used within 24"; 3+ is using Njal), Runes of Warding from Eldar which is board wide and increases chances of perils and Shadows in the Warp from Tyranids (cannot ally) which is only 12" and increases chances of perils. Both Rune Priests and Farseers are excellent options here as they can attempt to stop all types of psychic powers at a pretty consistent rate. If you're looking for allies from these armies, a good HQ choice is going to be these guys as you get this roll on top of your Deny the Witch roll.
In the end psychic defenses aren't going to be as reliable as they were before, particularly with the reduction in psyhic hoods effectiveness (which were the most common forms of psychic defenses before) but every squad at least has a chance of stopping psychic powers coming their way.
Psychic Power Generation -
Here is where we see a major difference from the Fantasy Magic system. Each Psyker rolls a set amount of dice on any number of tables. Unlike in Fantasy, most psykers only have two dice to throw at each table and if you get a double, you re-roll until you get a new result rather than choosing what you like. This means the tables and power generation are MUCH more random and you cannot rely on getting a specific power you want or need unless it is the primaris (any power can exchanged for the primaris of the given discipline).
This makes replacing your bog standard book powers with anything from the BYB quite risky unless the discipline you are taking from has a good primaris or a bunch of good powers you can use in any given situation. When you start hoping for a specific power or two and wishing you could cherry-pick, it's probably a good idea to head back to your bog-standard codex powers. The one good thing about this compared to Fantasy is you can take powers from multiple disciplines - just not from your codex and BYB unfortunately.
Types of Psychic Powers -
For older powers in current army rules - these types do not override what they are. It is "clearly expressed" within the specific army entry how they work. These new types of powers are meant to represent the new powers seen in the BYB and in future releases.
Blessing: Used at the start of the movement phase - targets friendly units and can affect units locked in combat and the Psyker itself and cannot take characteristics above 10 or below 1. Last a full game turn.
Conjuration: Used at the start of the movement phase and deep-strike a unit within range of the power.
Malediction: Used at the start of the movement phase and are basically the opposite of a Blessing (targets enemy units). Can affect units in close combat and cannot raise characteristics above 10 or lower them below 1.
Witchfire: The four types of powers below are all witchfires which are shooting attacks and used in place of firing a weapon in the shooting phase (unless the psyker can fire two, i.e. an MC psyker). These are basically psychic shooting attacks with an assault weapon profile unless noted otherwise and must roll to hit unless noted otherwise by its specific sub-type or is a blast/template.
Beam: Automatically hits all models under beam line. Beams reduce in strength as it hits more models by one strength per model - include ruins/walls in this classification as well. If the strength reaches 0, the beam stops. The first model hit is the part of the squad targeted by the rest of the psyker's squad (if any) and the beam can hit models out of LoS.
Focussed Witchfire: A sniping shooting psychic attack where on the psychic roll, five or less let's you snipe a model and on a six or more a randomly selected model in the unit is chosen. Not great reliability here for sniping but a nice bit of utility in terms of picking out special weapons, etc. Is particularly unlikely against something which forces you to roll more dice and keep them though a bit more reliable when rolling more dice and discarding the highest.
Maelstrom: Automatically hits all units (friendly and enemy) within power's maximum range regardless of normal restrictions.
Nova: Same as Maelstrom but always a shooting attack and only against enemy models.
With a bunch of new psychic disciplines, and some of them do have good primaris', and the continuation of good psykers from 5th edition, psychic might and defenses are going to be as important as ever. Their implementation is certainly different and as this is being written before any 6th edition books are released, one imagines we'll see a lot more of the different types of powers (i.e. Novas & Maelstroms) and I wouldn't be surprised at a general upward trend of psychic users.
However, a big limitation on all of this is the random power generation from the BYB tables and unless all the new army books have codex powers done like this, the codex powers will generally be the better options as you know exactly what you're going to get. Otherwise, any disciplines with good primaris in case you roll poorly are going to be much better choices to alleviate some of the randomness associated with rolling on such tables.