Much discussion about Mormonism revolves around whether Mormonism qualifies as a form of Christianity and whether the LDS Church is a Christian Church. To a large extent the answer depends on how you define 'Christianity'.
Under a broad definition, Mormonism can qualify as a form of Christianity, though perhaps only barely. Under almost any strict definition, though, it can't and this would mean that Mormons aren't Christians.
Why Care if Mormons are Christians?
It's legitimate to wonder why anyone would care one way or the other whether Mormons qualify as Christians, especially on a site like this. There are a number of reasons why the question and the answer matter, even for atheists.
First, the issue is important because both Mormons and orthodox Christians agree that it is important. Mormons often get offended when it's suggested that they aren't Christian (even though they deny that any other church is genuinely Christian) and orthodox Christians can get upset when Mormonism is classified as a form of Christianity. If they both agree that the issue is important, it may be because there are important things at stake.
Second, it's arguably a factual question. This is also something that both Mormons and orthodox Christians will agree on. It's not a disagreement about taste or personal preferences, but rather of facts: either Mormonism is or is not Christian; either Mormon beliefs are or are not in line with orthodox Christian theology.
Third, the answer will necessarily influence how we approach Mormonism in a variety of ways. Mormons invest a lot of effort into defending the idea that they are Christian, but if it turns out they aren't then that will affect how we receive other claims they make about their religion. If Mormonism isn't Christian, that affects how they are classified for demographic purposes and claims about how many Christians there are in America.
Mormonism as a Form of ChristianityTo ask if Mormonism qualifies as a form of Christianity is to ask, implicitly, what Christianity is and why anyone qualifies as Christian. Every denomination of Christianity has their own answer and many of those answers exclude a large swath of the rest of the people who call themselves Christian and who are generally recognized as Christian. We should avoid such pitfalls but using highly restrictive definitions.
The broadest possible definition of Christianity is that whoever calls themselves Christian is a Christian. That's useful in a day-to-day, live-and-let-live approach, but it risks reducing "Christian" to meaninglessness. It doesn't always make a lot of sense to accept such a claim without caring about the reasons upon which it is based.
Slightly better but still broad is to define 'Christianity' as any religion where Jesus plays a central or at least important role. This is a bit more helpful in distinguishing Christian from non-Christian belief systems, though not always. Under this definition Mormonism would easily qualify as Christian, but then again Islam could arguably qualify as well and that wouldn't be right.
In my opinion, the broadest definition of Christianity that is still reasonably meaningful but not sectarian is "a religion centered around the belief that Jesus is Christ (Messiah)." There's a lot of intuitive justification for this because it's hard to defend a system as "Christian" if there is no "Christ" involved. It also avoids sectarian prejudices because an awful lot of beliefs are compatible with Jesus being the Messiah.
Under this definition, Mormonism would once again qualify as Christian — but Islam wouldn't because Muslims believe that Jesus was a prophet of God, not the Messiah. So under a broad but reasonably meaningful definition, Mormonism is a form of Christianity.
However, this requires ignoring just about all beliefs that just about all historic forms of Christianity have considered important — including beliefs which Christians have agreed upon, even while killing each other over other doctrines. What's more, this definition of Christianity is not one which Mormons themselves would accept as valid. So using this definition to classify them as Christians does so, in a sense, in spite of Mormon beliefs rather than because of them.
Mormonism as a Non-Christian Religion
It would be unreasonable to ignore all of those ways in which Mormonism differs from traditional, orthodox, historic Christianity. It would also be unreasonable to pretend that those differences are trivial or at least no more important than all the ways in which historic forms of Christianity have differed from each other.
Catholic and Orthodox Christians have considered their differences over filioque and the role of the papacy to be very important. Protestant and Catholic Christians have considered their differences over "faith alone" and transubstantiation to be very important. Those differences pale, however, in comparison to their beliefs about the nature of God and humanity as compared to the doctrines of Mormonism.
In historic, orthodox Christianity there's nothing remotely like the Mormon doctrines that God was once a human being, that humans can become gods themselves, that there are a multitude of gods out there already, that the universe is eternal, that humans originated as eternal intelligences that then became spirit children born from God and his wife (or wives) and then later born in physical form, etc.
That there is a relationship between Mormonism and Christianity is of course undeniable. Then again, there is also a relationship between Christianity and Judaism — and arguably a similar one. Christianity started as a form of Judaism, but quickly diverged too much from historic forms of Judaism to remain, so ended up becoming its own, independent religion with new, additional scriptures.
If you define Judaism broadly enough, you might be able to argue that Christianity is still Jewish. At least some Christian churches may arguably be as Jewish as the most liberal Reformed Judaism congregations. But however close the two systems may remain, they have nevertheless grown to be separate from each other.
It's reasonable to view the same as being the case for Mormonism and Christianity, and without relying on sectarian prejudices against Mormonism. It certainly started out as a form of Christianity and it clearly owes a great deal to Christianity, but it has arguably diverged enough, and did so early enough, to be considered its own, separate religious system.