acceptance is originally defined as 
1) receiving something that's arrived and offered, for example, accepting a package.
2) believing in something as true, for example Christians accepting Jesus as savior, or accepting the validity of an election, etc.

As a psychological activity wherein a person experiences a set of conditions—often uncomfortable and unwanted—without trying to change or avoid it. We're often told that acceptance is a central part of buddhist spirituality.

There is no direct word for psychological acceptance, as just defined, in pali. 

The buddha's never taught that we should experience suffering without trying to change it. All of the buddha's teachings are goal seeking, towards inner peace.

The buddha did teach about learning to develop contentment with whatever external circumstances we find ourselves in:
—santutthi, or developing contentment with our acquisition of four requisites, food, clothing, shelter, and medicine. Acquiring these requisites is not the ultimate objective of spiritual practice. The four requisites are simply the foundation upon which our spiritual, inner practice rests.
—this is a concept to be addressed in another talk. 

one of the buddha's key concepts that has much in common with acceptance is equanimity (or uppekha), which plays many roles in the path and has several meanings
—to the degree that equanimity means accepting things as they are and not trying to improve them, its only external conditions
—spiritually, the buddha's equanimity means selecting the areas in life that can be effectively improved and putting aside the areas where our efforts will have minimal, if any, effect.

equanimity based on multiplicity, the changing external conditions
—if we become attached to praise, fame, gain, we'll experience the downside of those states; blame; loneliness; loss. change is inherent.
— the principle of not allowing the mind to get carried away by good things or depressed by bad things; 
—its keeping the knowledge of pain & loss in mind when we're experiencing pleasure and gain, and vice versa.
—we don't become so thrown. we enjoy external things, but with the understanding they will pass.
—"this too shall pass" is a part of virtually all cultures and spiritual outloks; its not that original an insight.

this kind of equanimity requires keeping the mind from being swallowed up by external events, understanding that external conditions by nature change; that the happiness they provide isn't lasting or even generally blameless.

this type of equanimity is the lowest form of equanimity; when practiced alone it encourages a sense of fatalism and complacency. 

social equanimity: based on the  reflection on external karma 
this kind of equanimity balaces out divine attitudes of generosity, compassion, appreciation.
—that there are some areas in our social life we can't make changes
—people who seem happy even when their actions are evil
—good people who suffer and we cannot help.

in social equanimity it doesn't mean we sit passively, it means we focus our efforts on people we can help, to the degree that our efforts don't interfere with our inner tranquility.
—even if we find someone we can help. we don't become so involved in their life that we toss away our own inner peace.

so social equanimity means seeing who we can help, who we can't, and only getting involved to a practical degree.

equanimity based on inner karma
this means putting your mind in a position where it’s able to see and learn about cause and effect
—we have a mature relationship with the goal. sometimes our past karma will be too strong and there'll be so much stress that it wont be possible to relax certain parts of the body (example the shoulders or stomach). a sense of balance means we acknowledge this but still aim to relax what we can.
—sometimes certain voices in the mind will be too loud, and it'll be difficult to replace them; there are other strategies, like labelling the thoughts, welcoming, etc.
—when things go well, we learn why they’re going well. we don't get carried away by how great we feel, we inquire, what is causing this state?

equanimity through karma is not having preferences. 
—we don't let things that feel good in the short term be automatically equated with skillfulness.
—don't let our habits or "stuff that defines us and gives us individuality" be seen as a given or inevitable.
—some people prefer a path that's just accepting, or letting go, without having to put effort into virtue, concentration practice, generosity. but our preferences have nothing to do with what we need to develop.
—ajahn maha boowa said that the effort peace requires wont kill us. its just more than we often want to put into our practice.

we don't just settle for any body or breath sensations or thoughts, we learn which kinds of breathing and body states and thoughts are possible right now, cultivating the greatest amount of peace and tranquility

so this equanimity is based on acknowledging what types of concentration practices are possible right now, which aren't, and not having preferences to what works and what doesn't work.

unconditional equanimity
as we settle down the mind skillfully, the mind becomes full with whatever we're observing (perhaps the breath).  a sense of refreshment and ease fills the body. eventually there's a sense of equanimity in mind; its not a worldly pleasure; its a pleasure that has no drawbacks; it is an balance and evenness between effort and ease.
—still, when we reach this kind of equanimity there's always more to do; equanimity isn't synonymous with nibbana; its a way station on the path to it. 

finally when we reach great levels of peace and inner concentration, a not needing to get involved with perception, it takes care of itself. but this level of equanimity takes care of itself. its not created or worked on.