People have occasionally questioned the authenticity of the work, particularly of the last book, on the grounds that it seems to have more rhetoric and less of Xenophontic simplicity than other works by Xenophon; but such suggestions have tended to be in the minority. The first two books are more or less excerpts from Xenophon's Hellenica (Xenophon's history of the last part of the Peloponnesian War and its aftermath).
You can read the Agesilaus online in Bowersock's translation at the Perseus Project and in Dakyns's translation at Adelaide.
Agesilaus II (c. 440 - c. 360) was an unusual Spartan king. It was not originally expected that he would ever be king. He was born the younger son of Archidamus II, and, moreover, he was born lame, with one leg shorter than the other. He was short and physically unimpressive. But coming to the throne by happenstance at a crucial time, he became highly popular among the Spartans and extremely powerful throughout Greece. Xenophon knew him personally and served under him for several campaigns.
Xenophon gives his reason for writing the work right at the beginning:
I know how difficult it is to write an appreciation of Agesilaus that shall be worthy of his virtue and glory. Nevertheless the attempt must be made. For it would not be seemly that so good a man, just because of his perfection, should receive no tributes of praise, however inadequate.
Thus there is no pretension of evenhandedness here; the point is to praise Agesilaus for his virtues and his achievements, because he was a good man. And we begin with context -- he is a man from an excellent family in an excellent city, chosen by that city to be its king. having recently become king, Agesilaus is faced with a challenge: Persia was preparing to invade Greek lands again. Agesilaus proposed a counter-invasion to nip this in the bud, and the proposal was enthusiastically accepted, particularly as it would mean that Persia, for once, would be the scene of the Persian-Greek showdown.
The Persian governor, Tissaphernes, tried to hold Agesilaus with proposal of an armistice until he could get reinforcements, but Agesilaus was pleased to discover the offer fraudulent -- it put his invasion entirely in the right. Tissaphernes expected that Agesilaus would build a vengeance strike against himself, and prepared accordingly, but in fact Agesilaus continued to be levelheaded and instead worked on systematic conquest of wealthy and strategic cities. Out of this conquered wealth, Agesilaus richly rewarded all of his friends, which, of course, led to others working very hard to be his friends. But Agesilaus, taking the long view, also recognized that these lands needed to provide not only the temporary wealth of looting but a consistent and easily held supply base. For this reason he actively treated prisoners of war well and made sure, whenever he conquered a city, that he could claim to have liberated it, by making penalties less harsh and the laws more free. In order to build a cavalry, which was essential for the terrain, he obligated the horse-breeders to supply it, and he set in place a policy freeing from obligation to serve in the cavalry any wealthy horse-breeder who would provide a rider and arms. This made powerful locals active supporters of his cavalry-building. Meanwhile he trained his army, guided by the three principles Xenophon says provide for military success: reverence to the gods, active practice as a military, and obedience to command. And success Agesilaus certainly had before the Spartans called him back to deal with an anti-Spartan confederation of Greek cities.
Xenophon has no patience for people who praise generals for fighting forces greater than their own -- while it may sometimes be necessary, only an idiot does it freely and deliberately. Agesilaus comes against the confederates with a large and well trained army with considerable morale and military pride from their prior fighting. Xenophon thinks that Agesilaus could have done better tactically, but he nonetheless still showed valor and tactical care, guaranteeing victory to his army.
Even a competent king cannot eliminate all possibility of disasters, and Xenophon seeks to absolve Agesilaus of any blame in less fortunate happenings, such as a new anti-Spartan confederation or slave revolts. By this point, Agesilaus was aging rather severely and no longer able to take an active part in fighting, so he turned his mind to the second most important thing Sparta needed for her protection: money, which he began to gather by diplomatic means from both allies and defeated foes.
* Xenophon's praise of Sparta for unbroken continuity in its line of kings is not a minor note -- stability of government is something Xenophon consistently regards as one of the most important questions of politics throughout his work.
* The claim that Agesilaus was still a youth when he became a king has baffled scholars through the ages -- Agesilaus would have been in his forties by that time, and Xenophon could hardly have been ignorant of that fact. Some have suggested that a 'not' had dropped out at some point; then the passage might be read as suggesting that Agesilaus, although not a youth and still new to politics, took admirable initiative in a time of crisis.
* Books I and II are basically background, dealing with external matters. Xenophon's primary interest, however, is virtue, and it is to this that he turns in Book III.