Certain Eidolon types obtain energy resistance against some type of damage - for example the Psychopomp type gains cold resistance 10 and electricity resistance 10 at 4th level - while some obtain the Resistance evolution - for example the Angel type obtains the resistance (acid) and resistance (cold) evolutions at creation - but what is not clear to me is this: does the Resistance evolution stack with the first (Psychopomp's) energy resistance? Or does it stack with nothing? I'm asking because while it's clear that it's not meant to stack with itself, it is a little more nebulous with the first case.
In the 2011 Paizo messageboard thread "Do energy resistances stack?" creative director James Jacobs and another user have the following exchange:
A tiefling begins with an [e]lectricity resistance of 5. When he becomes a level 3 [i]nfernal bloodline [s]orcerer, he gains an electricity resistance of 5. Does that stack up to 10 or does it just stay at 5?
Nope. Energy resistance does not stack.
While Jacobs claims not to be a rules guy and his rulings are his own, his pronouncements are often taken by fans as indications of how Paizo imagines the game to be played. (Jacobs was also a designer on Dungeons & Dragons, Third Edition—one war story is here, for instance—that was later updated by the 3.5 revision. It's the 3.5 revision that forms Pathfinder's spine. In other words, he kind of knows his stuff, too, although it takes Pathfinder lead designer Jason Bulhman to make absolutely official pronouncements about the rules.)
If a more mechanical explanation is necessary, probably the easiest way of viewing this issue is that a creature that gains energy resistance to the same kind of energy from two different sources gains two different abilities, yet those abilities do the same thing. Thus the two abilities don't stack but overlap, and each time that ability's effect arises, the creature picks which to use. (Obviously, it often picks the bigger or better ability.)
With that in mind, it may be helpful to compare getting energy resistance to the same energy type multiple times to getting a fly speed multiple times. For example, an ancient red dragon has a fly speed of 250 ft. If it casts the spell fly, its fly speed doesn't become 310 ft. Instead, the dragon has a fly speed of 250 ft. and another different fly speed of 60 ft. When it flies, the dragon picks which of its fly speeds to use. (By the way, a dragon might do this as a precaution: the fly speed from the fly spell allows the dragon to fly even if it can't flap its wings. Also, the dragon's natural fly speed has a maneuverability of clumsy while its fly speed from the spell has a maneuverability of good; see also the skill Fly.)
Similarly, an unchained eidolon that possesses the subtype angel does "gain electricity resistance 10 and fire resistance 10" when its summoner is level 4, and one that possesses the subtype psychopomp does "gain cold resistance 10 and electricity resistance 10" when its summoner is level 4, but these resistances don't combine with the 1-point evolution resistance that says
The eidolon gains resistance 5 against that energy type. This resistance increases by 5 for every 5 levels the summoner possesses, to a maximum of 15 at 10th level. This evolution can be selected more than once. Its effects do not stack. (Emphasis mine.)
While a reader may intuitively parse that boldfaced sentence as Its effects do not stack with itself (e.g. having the resistance evolution resistance three times before level 5 and picking cold each time does not provide cold resistance 15), that sentence can actually be read far more expansively: Its effects do not stack with anything ever unless a specific exception says so.
However, what that energy resistance can do is improve if the eidolon somehow acquires one of the rare abilities that improves energy resistance rather than simply granting another iteration of it. For example, the feats Energized Wild Shape and Undead Companion make it clear that they can, in fact, actually increase a creature's existing energy resistance, while the feats Efreeti Stance and Fiendish Energy Resistance just offer other iterations of energy resistance that don't improve a creature's existing energy resistance.
Note: I wish the game were much more up front about this, the rules somewhere saying outright what Jacobs says in that thread referenced above. It seems that Pathfinder just kind of expects readers to reach this conclusion on their own rather than the game itself actually saying it.
Unless otherwise specified resistances to the same element do not stack, and you simply use the highest resistance to that element. This is a general rule that still applies in this situation.
Stacking refers to the act of adding together bonuses or penalties that apply to one particular check or statistic. Generally speaking, most bonuses of the same type do not stack. Instead, only the highest bonus applies. Most penalties do stack, meaning that their values are added together. Penalties and bonuses generally stack with one another, meaning that the penalties might negate or exceed part or all of the bonuses, and vice versa.
Note that although arguably the rules on bonuses do not exactly apply here, as resistance and damage reduction may not be bonuses or penalties (depending on how you define 'checks or statistical scores'; arguably they are a bonus or penalty in the same way bonuses to damage are counted a bonus). However the rules for energy resistance specify that “Each resistance ability is defined by what energy type it resists and how many points of damage are resisted.” - thus the energy type could be considered synonymous with bonus type. I.e. fire resistance 5 stacks with cold resistance 10, just as an armour bonus of 1 stacks with a natural armour bonus of 2.
Similarly to the above sources the rules on Damage Reduction are very clear that multiple damage reductions do not stack, and give an example of how the general rule above applies to a situation they arguably don't explicitly cover.
If a creature has damage reduction from more than one source, the two forms of damage reduction do not stack. Instead, the creature gets the benefit of the best damage reduction in a given situation.