This question comes up from time to time, so I’ll answer as best I can. I choose to use the definition in its pure, unadulterated form, a close approximation of which is given by Investopedia:
A financial benefit that is realized when the amount of revenue gained from a business activity exceeds the expenses … needed to sustain the activity. [link]
This is quite close to the definition I recently used in Profit IS Justified by Entrepreneurial Risk: for any given interval, Profit equals Revenue less Expenditures. Incidentally, this is not that far off from Franc’s definition (you may recall that Franc vehemently disagreed with the position I took in the aforementioned post) given in a post about the Corporate profit motive, with which I agree at least somewhat  in spirit:
In a free society, where there is no exploitation, the profit you make on a given product is, in general, the material expression of how other people evaluate the worth of your product… individual profit [is] something we can understand and contend with in a free society.
You’d almost think he was channeling Mises. Although individuals may also experience “profits” by exploiting others (more on this, below), there is nothing either explicit or implicit in this definition about exploitation. Consequently, profit (or loss) is the end-result of all production, undertaken by anyone, within any socioeconomic framework. Profit (or loss) is what remains after the bills are paid. Therefore I conclude that exploitation is neither a necessary nor sufficient condition to bring about profit, and I reject the Trojan Horse definition, which smuggles the assumption that profit requires exploitation.
But what about the exploitation? If your definition of “profit” includes this smuggled premise, in my opinion you are really describing an entirely different animal. I more closely associate the exploitative phenomena with economic rent (The Economist, for its faults, gives a very precise textbook definition):
[Economic rent] is a measure of market power: the difference between what a factor of production is paid and how much it would need to be paid to remain in its current use… In perfect competition, there are no economic rents, as new firms enter a market and compete until prices fall and all rent is eliminated.
I’ve alluded to this in the past. The invaluable functions of the price system (which have no corollary in a non-market economy), are its ability to send signals  to the market. By following profits and avoiding (or cutting) losses, the tendency of a free economy (one where competition is as atomistic as possible) is towards equilibrium, and the existence of profit is a fleeting disequilibrium which signals to individuals, “Not enough of X is being produced!”
For if enough were being produced, there would be no profits to be had, and if instead too much were being produced, there would be mounting losses.
Production will increase in those spheres where profit is to be gained, the increasingly available quantity of product in this market will drive prices down to a theoretical minimum or perhaps temporarily even lower. At this “equilibrium” price revenues are just enough to satisfy opportunity costs. Accordingly, the experiences of “profit” or “loss” are the calculus used to right the economy when it’s out-of-whack — to coordinate the infinite array of human desires with the finite array of resources at our disposal.
At the very least then, the exploitation theory of profit would need to assert that non-exploitative profit is a catalyst for economic rents, in other words, it would need to negate everything we understand about perfect competition and economic disequilibria, and most of price theory, to boot.
The “evenly rotating economy”, one with neither profit nor loss, (in which all needs are omnisciently forecast, and all resources are allocated in precisely the right quantities, times, and places, such as to satisfy those needs at the precise moments of their realizations) does not exist.
If, under the above definitions of profit, rent and competition, you’re still asking yourself why anyone needs to profit, you might as well be asking yourself, “Why does anyone need to eat?” As the purpose of eating is to sustain life, the purpose of production (whether it be for your own use or another’s) is profit. Eating permits us to live another day, while profit allows for that day to perhaps be more enjoyable than the previous.
- c.f. Mike Gogulski’s reply for an accurate summarization of my objections thereto.
- c.f. Mises 1949, p.299. “Production for profit is necessarily production for use, as profits can only be earned by providing consumers with those things they most urgently want to use.” — h/t QuebecoisLibre
- Prices convey information about the magnitude and intensity of any of the infinite array of human wants, desires, and needs, and about how to best allocate the finite resources at man’s disposal, which can in no other way be conveyed as quickly, as precisely or with as little effort.