The Mask of the Betrayer has a totally revamped influence system which can now provide gameplay benefits to your companions (and yourself in some cases) when they gain sufficient loyalty to you.
The depth of the companions has also been increased by having them also react more believably to poor treatment. If you upset them too much, they will also act appropriately. At first, you'll lose direct control of them (we've improved upon the companion artificial intelligence). Eventually, they'll stop sharing items or carrying around your loot for you. And if you oppose them enough, they'll abandon you. Don't worry, if they annoy you too much, you can kill them, which might bring some benefits.
Obsidian has also overhauled the influence system that governs your companion's reactions to you. You can actually check a numerical influence rating on your companion's character sheets; raising this number will allow both you and your companion to gain power. A high influence with a companion will let them give you bonus feats, and you'll return the favor. For example, as you gain more influence with Kaelyn, it would eventually unlock new talents that you can only get through her that were probably a bit more geared to defense, while One of Many had a handful of dark powers to imbue upon you, should you prove that you're worth his trouble. The only problem is that you can't possibly get a solid reputation with both, as when you gain influence with one, it goes down with the other. Eventually, if you gain too much influence with one, it won't sit well with the other, and they'll eventually leave the party. If you really upset a companion, they may even turn on you and attack, forcing you to battle a formidable opponent.
If you like playing in an especially hardcore fashion, though, you'll be pleased to know that all of the companions are optional in the expansion; you'll never be required to take anyone along on your adventures if you don't wish to.
In addition, you no longer need to resort to the console to check your influence levels with your companions. Simply opening up their charcter sheet will enable you to see how much influence you've gained or lost with them.
Benefits of Gaining Influence
Each companion gives you two stages of power based on influence - a minor enhancement once you reach a certain influence level (50+), and a superior version of that enhancement once you reach a higher influence level (75+). A third level of benefit is granted if you choose to pursue a romance with an available companion; Gann for females and Safiya for Males. These enhancements are typically boosts to a stat based on the class of the character granting them, but they can also include additional benefits. These stat boosts will stack with stat boosts gained from items. In addition, as you gain influence your companions can also gain up to 3 benefits from you, typically starting at an influence of 25+, plus an additional benefit if you pursue a romance with the appropriate character. Make sure that you talk to your companions often. For more details check out the Influence Benefits page.
The following is taken from the influence guide posted by "Dexter111" on the Bioware forums. We have added a little text and have converted his coded influence into real in-game influence. As time evolves we'll add to this and make necessary changes and revisions, but we don't expect those to be numerous or significant. To make the influence guide easier we've broken it up into three sections and set them off on separate pages. Simply click on the any of the three parts of the guide or the internal part of the guide and you will jump you to the appropriate point in the game.