In Buddhism what is 'the will'?

"The will" is any kind of desiring, wanting, thinking, intending, choosing, doing, ... Be it very very subtle, very subtle, not very subtle, strong, very strong, etc.

Yes, "the will" comes from greed, hatred and delusion. All come from delusion. When all delusion is removed, "the will" is extinguished.

In the same way as a flame is extinguished when all the fuel is removed, "the will" is extinguished when all delusion is removed.

By removing all delusion, "the will" remains until our bodies die, but because delusion is removed, "the will" is known, it's cause is known and it's ending is known.

"Knowing" leads to final liberation: complete extinguishing of all will.

There's a sutta called AN 11.2 Cetana Sutta, which Thanissaro Bhikkhu subtitled "An Act of Will".

The Cetana Sutta is similar to AN 10.1, which is quoted as The rewards of virtue.

Anyway, the Cetana Sutta starts with...

For a person endowed with virtue, consummate in virtue, there is no need for an act of will, 'May freedom from remorse arise in me.' It is in the nature of things that freedom from remorse arises in a person endowed with virtue, consummate in virtue.

... and continues in a similar way.

I read that as saying that being "consummate in virtue" is sufficient: that leads to freedom from remorse, to joy, to etc., through to dispassion, and to knowledge & vision of release.

Perhaps this is similar to Andrei's answer which says:

The key to exertion is not as much forcing oneself, as it is an ability to derive delight from the element of doing the right thing.

It also seems to match other answers in identifying 'will' as 'cetana'.

My personal guess/opinion/interpretation of how it works as described in the sutta is an internal monologue that goes something like,

• If I do the wrong thing or don't do the right thing then I'll regret it (i.e. feel remorse) which is unpleasant/regrettable
• I don't want to feel remorse (or don't want to continue to feel remorse)
• i.e. what I do want instead of remorse is that "may freedom from remorse arise in me"
• Having desired freedom of remorse, and having seen (right view) that what leads to freedom from remorse is virtue (including Right Speech, Right Action, and Right Livelihood), then I (naturally want to) do the right thing in order to attain the desired result.

Saying "there is no need for an act of will" reminds me of Wu wei -- a way in which I interpret that is that if I want to score a goal in football then it doesn't take an act of will to kick the ball ... instead kicking the ball follows naturally from being a good football player, from the desire to play football and to score a goal.

Another interpretation is based on dependent arising: when that when virtue is present then that's the condition for no-remorse to arise naturally (like soil and seed and nutrients and water and sun is the condition for a plant to arise naturally).

Lastly, Buddhism doctrine talks about fetters and hindrances and so on. So what an English-speaker might want to characterize as "an absence of will" (e.g. "I don't want to get up in the morning"), a Buddhist might characterize as the presence of hindrances e.g. of sloth and torpor (but see also Hui-neng's mirror).

Is there a Buddhist concept that maps to the western concept of the will? For instance the will to get up on a morning, to go to work, practice Buddhism etc...

There's the Four Right Kinds of Striving (Sammappadhana), part of the 37 Aids to Enlightenment:

MN 77: “Again, Udāyin, I have proclaimed to my disciples the way to develop the four right kinds of striving.

1. Here a bhikkhu awakens zeal for the non-arising of unarisen evil unwholesome states, and he makes effort, arouses energy, exerts his mind, and strives.

2. He awakens zeal for the abandoning of arisen evil unwholesome states...

3. He awakens zeal for the arising of unarisen wholesome states…

4. He awakens zeal for the continuance, non-disappearance, strengthening, increase, and fulfilment by development of arisen wholesome states, and he makes effort, arouses energy, exerts his mind, and strives. And thereby many disciples of mine abide having reached the consummation and perfection of direct knowledge.

Will is cetana, 'consciousness, sense, thought, intention,' the original and originating kinetic aspect of trans-dual reality. Intention is of course the essence of karma, correlative to the "act of observation" in quantum physics.

It is the root of samsara, the principle where samsara originates in reality, not in the sense of a temporal sequence but in the sense of logical primacy. Desirous attachment is will in its objectifying mode. Based on your examples it might also be correlated with 'energy' (viriya'), to which the Buddha frequently exhorts his followers, contrary to the view that Buddhist practitioners are "passive."

The will to do something would probably be equivalent to the term cetana, which means intention or an act of will. The term itself doesn't have any particular positive or negative connotation, as you can have a cetana that is good just as easily as a cetana to do something bad.

Most significantly, the cetana formed with an act is what determines the kind of karma that is produced, as is explained in AN 6.063 where the Buddha said:

Intention, I tell you, is kamma. Intending, one does kamma by way of body, speech, & intellect.