What is the archaeological evidence for the events in the Book of Mormon?

There is no archeological evidence that can be said to prove the book of Mormon, in the same way, that there is in the Bible; because the Mormon church does not allow for any real investigation of the claims of the BOM.

The most glaring example of the prohibition of archaeological investigation is at the site of the Hill Cumorah, where Joseph Smith supposedly found the gold plates which were translated into the BOM, and where according to the BOM there was a massive battle of extermination (which if true, would have been the largest battle in recorded history, certainly worth investigating for merely scientific reasons unrelated to the BOM), where there were supposedly millions of combatants, who were all destroyed, in upstate New York. Imagine millions of bodies, millions of swords, spears, and arrows. An event this massive and important would surely be one of the most important moments in human history.

The Mormon church (who bought the land) however, has never allowed, and will not allow the site to be excavated, therefore there is no evidence whatsoever that any of these events actually happened as described in the BOM.

"The hill, which was known by one division of the ancient peoples as Cumorah, by another as Ramah, is situated near Palmyra in the State of New York." (James E. Talmage - Mormon apostle, Articles of Faith, Lecture 14, p.262)

"The great and last battle, in which several hundred thousand Nephites perished was on the hill Cumorah, the same hill from which the plates were taken by Joseph Smith." (Orson Pratt - Mormon apostle, Journal of Discourses 14:331)

"Finally, they became so utterly wicked, so fully ripened for destruction, that one branch of the "nation, called the Nephites, gathered their entire people around the hill Cumorah, in the State of New York, in Ontario County; and the Lamanites, the opposite army, gathered by millions in the same region." (Orson Pratt - Mormon apostle, Journal of Discourses 17:30)

"Thirty-six years prior to this time his nation was destroyed in what we term the State of New York, around about a hill, called by that people the Hill of Cumorah, when many hundreds of thousands of the Nephites—men, women and children, fell, during the greatest battle that they had had with the Lamanites." (Orson Pratt - Mormon apostle, Journal of Discourses 20:63)

The lack of evidence proving (or definitively disproving) the BOM is not because the BOM does not provide enough detail to excavate (as was stated in another answer), it is because the Mormons will not allow their faith to be tested or examined scientifically or logically or historically, in the same way that the Bible and Christianity (and every other religion) is constantly being criticized and examined by both those from within and outside of it.

Depending on the scope of this question, it can be very difficult to give any satisfactory answer to. Archaeology primarily concerns itself with artifacts that have been left behind from ancient days, whereas the Book of Mormon deliberately avoids going into too much detail on the culture of its people and explaining details that might aid archaeological research, preferring to focus on more sacred matters. Also, it appears that their preferred method of construction was in wood, which doesn't lend itself well to the generation of archaeological ruins:

From Helaman chapter 3:

3 And it came to pass in the forty and sixth [year], ... there were an exceedingly great many who departed out of the land of Zarahemla, and went forth unto the land northward to inherit the land.

...

7 And there being but little timber upon the face of the land, nevertheless the people who went forth became exceedingly expert in the working of cement; therefore they did build houses of cement, in the which they did dwell.

...

9 And the people who were in the land northward did dwell in tents, and in houses of cement, and they did suffer whatsoever tree should spring up upon the face of the land that it should grow up, that in time they might have timber to build their houses, yea, their cities, and their temples, and their synagogues, and their sanctuaries, and all manner of their buildings.

10 And it came to pass as timber was exceedingly scarce in the land northward, they did send forth much by the way of shipping.

11 And thus they did enable the people in the land northward that they might build many cities, both of wood and of cement.

So we see that they considered timber to be the preferred method of construction. When it was unavailable, they developed new techniques for constructing houses and buildings of cement, and eventually became "exceedingly expert in the working of cement," but still placed a high value on wood as a construction material. (Having lived in South America, where many modern homes are built of cement and bricks, I can certainly understand this attitude. Cement is a rather good conductor of heat, which is an undesirable quality at low latitudes!)

The cement settlements are interesting. This was once derided as a hopeless anachronism in the Book of Mormon: the working of cement was an Old World art that never existed in pre-Colombian America! Right up until the mid-twentieth century, that is, when archaeologists started finding cement settlements in the vicinity of modern-day Mexico City, built with an extremely high degree of skill ("exceedingly expert") dating back to approximately the same time as the Book of Mormon states that these colonists moved in. (As reported in Concrete Evidence for the Book of Mormon.)

However, this settlement is an exception rather than a general rule. The people in question preferred construction in wood or simply living in tents, neither of which generates much in the way of ruins. Also, Book of Mormon archaeology is at a second disadvantage when compared to Biblical archaeology: on the Biblical side, we know exactly where to start looking! Modern-day Jerusalem is located on the side of the same mountain it's been on since at least the days of Melchizedek, to give just one example.

It will not be surprising, if you understand this, to hear that much of the truly interesting archaeological and historical evidence to arise in support of the Book of Mormon actually comes from discoveries in the Old World. To continue the earlier theme of Book of Mormon details once thought to be ridiculous, the book mentions two different men by the name of Alma. This was once held up as proof of Joseph Smith's hopeless inexperience: Alma is a Latin name, not a Jewish name, and a feminine Latin name at that!

...until the Dead Sea Scrolls were discovered, more than a century after the publication of the Book of Mormon, that is. One of the scrolls contains a reference to one "Alma, son of Judah." Oops.

Then we have the Lachish Letters, a group of clay tablets found in the ruins of an ancient Jewish city, dating back to just a few years after the start of the Book of Mormon. They describe several non-Biblical details of the local political climate which fit hand-in-hand with the corresponding Book of Mormon narrative. LDS scholar Hugh Nibley wrote a very detailed article on the letters, summarizing it at the end with 18 points of remarkable similarity between the two, and concluding that

Other parallels may be added to taste, but this should be enough to show that Joseph Smith was either extravagantly lucky in the opening episodes of his Book of Mormon—that should be demonstrated by computer—or else he had help from someone who knew a great deal.

(Two Shots in the Dark, first half)

Entire volumes have been written, and then, sadly, largely ignored, regarding the authenticity of the introductory chapters of the Book of Mormon, the ones that deal with travels in the Old World. To give just one example which is freely available online, see Lehi in the Desert, again by Dr. Nibley. He examines cultural details, religious and dream imagery, historical background, and even the names used, and shows that it is all remarkably consistent with the time and place it purports to originate from.

Stronger than any external evidence for or against the authenticity of the Book of Mormon, though, is internal evidence. There is a word for a literary work that claims a certain authorship but does not actually originate the way it claims to: forgery. And there are well-established tools and techniques for the detection of literary forgery, and they do not require the support of external evidence. But strangely enough, these formal methods don't tend to get applied in adversarial analyses of the Book of Mormon.

On the surface of it, there's no good reason not to; the Book of Mormon is a forger's nightmare! A fairly large work, claiming to be historical in nature, disseminated as widely as possible and inviting all to read and critically examine it? What is the forger thinking?!? This violates every rule of producing a successful forgery! But the two most important principles in the detection of literary forger tend to be consistently ignored:

• Begin with the assumption that the work is genuine, and
• no external evidence is required to expose a forgery of any significant size; internal inconsistencies alone will do that every time.

When the formal, scientific techniques for detecting forgery are applied to the Book of Mormon, instead of emotionally-driven attempts to reach a predetermined conclusion, the book is shown to be genuine. (Dr. Hugh Nibley, New Approaches to Book of Mormon Study)

I could go on, (and on and on and on; as I said before, there have been entire volumes written on the subject,) but this answer is getting to be too long as it is. Suffice it to say that a large corpus of evidence exists which, while not enough to conclusively prove the truthfulness of the Book of Mormon, (they haven't even managed to do that with the Bible, so it would be unfair to hold another work that claims to be scripture to such a high standard,) definitely demonstrates that the book is worthy of serious attention and consideration.