Is "Mormon" a disrespectful term?

In a 2018 address to the worldwide church, President Russell M. Nelson addressed this topic with direct and unequivocal language. In the address, he quotes the revelation given to Joseph Smith that the church should be called The Church of Jesus Christ of Latter-day Saints, and that omitting Christ's name when referring to the church or its members takes the focus away from Christ.

What’s in a name or, in this case, a nickname? When it comes to nicknames of the Church, such as the “LDS Church,” the “Mormon Church,” or the “Church of the Latter-day Saints,” the most important thing in those names is the absence of the Savior’s name. To remove the Lord’s name from the Lord’s Church is a major victory for Satan. When we discard the Savior’s name, we are subtly disregarding all that Jesus Christ did for us—even His Atonement.

He goes on to encourage church members not only to avoid "Mormon" when referring to the church (as has been done before), but also when referring to its members.

If someone should ask, “Are you a Mormon?” you could reply, “If you are asking if I am a member of The Church of Jesus Christ of Latter-day Saints, yes, I am!”

If someone asks, “Are you a Latter-day Saint?” you might respond, “Yes, I am. I believe in Jesus Christ and am a member of His restored Church.”

The currently accepted answer accurately describes the origins of the term and its use, and other answers provide additional useful context. But while using "Mormon" is still unlikely to cause offense, the church and its members—since President Nelson's recent address—have been making a deliberate effort to include Jesus Christ's name to an extent that is unprecedented in the history of the modern church.

Just to perhaps clarify one point in Mason Wheeler's answer:

The word "Mormon" comes from the title of LDS primary scripture, the "Book of Mormon". "Mormon" is the name of a prophet who is a central character in that book.

I'm tempted to say that the fact that the word is the name of one of their faith's leading prophets should mean that there would be no reason for LDS folks to find it offensive. But I suppose that isn't necessarily so. What makes one word for something offensive and another not is a mysterious and complex thing.

The word Mormon comes from the Book of Mormon which is named because one of the characters. A prophet named Mormon took many records and abridged them to make what we have today as the book of Mormon. The official name is the Church of Jesus Christ of Latter Day Saints. The only real reason the word Mormon is discouraged is because there is confusion on whether Mormons believe in Jesus. The use of the full name quickly ends the confusion. It is now so common to refer to member of the Church of Jesus Christ of Latter Day Saints as Mormons, it is now being used by the church in order to stop confusion as to who is being talked about.

I am an LDS Missionary and often I will refer to myself by the full name of the church, then I will follow up by clarifying that we are Mormon. Most people don't know the full name so it ends confusion to just use the term everybody uses. It isn't offensive, just discouraged so people can learn the official name of the church. Further clarification declaring ourselves as Mormons isn't a problem.

This topic has already been very thoroughly answered, but I just found this in the LDS Church Handbook which seems instructive, so I thought I'd quote it here:

Section 21.1.34

Referring to the Church and Its Members

As the Church grows across boundaries, cultures, and languages, the use of its revealed name, The Church of Jesus Christ of Latter-day Saints (see D&C 115:4), is increasingly important in the responsibility of the Church and its members to proclaim the name of the Savior throughout all the world. Accordingly, references to the Church should include its full name whenever possible. Following an initial reference to the full name of the Church, the contractions “the Church” or “the Church of Jesus Christ” are acceptable.

Referring to the Church as “the Mormon Church,” “the Latter-day Saints Church,” or “the LDS Church” is discouraged.

When referring to Church members, it is preferable to use the phrase “members of The Church of Jesus Christ of Latter-day Saints.” As a shortened reference, “Latter-day Saints” is preferred and “Mormons” is acceptable.

The word Mormon will continue to be used in proper names like the Book of Mormon and the Mormon Tabernacle Choir. It will also continue to be used as an adjective in phrases such as “Mormon pioneers.” In addition, it may be necessary to use the word Mormon to identify the Church as it is commonly known in some countries.

Interestingly, the use of the term "LDS Church" is discouraged, and on some occasions, the term "Mormon ..." is quite acceptable.

I know a very prominent Mormon family in Washington state (where I live). The father is a county Precinct Committee Officer for the Republican party and an Usher at their church. I am good friends with them, and have spoken to them about their faith a few times. In my conversations with them, they indicated that the term is neither offensive or condescending if used in the proper context. If there is a cynical tone to a phase such as "oh, he's a Mormon," then of course it could be offensive. If, however, you are merely using the term as a matter-of-fact to easily identify their religious preference, it is not construed as a negative connotation.

Elder M. Russell Ballard of the Quorum of the Twelve Apostles, recently spoke of the use of "Mormon" vs the full name of the church, The Church of Jesus Christ of Latter-Day Saints.

See the Article: We can all be more consistently involved in missionary work by replacing our fear with real faith.

It is correct to use the term "mormon" except when referring to the church itself. It is the church of Jesus Christ of Latter-Day Saints, not "Mormon".

UPDATE: here is the full quote from the above link.

In the October 2011 conference, I urged that we remember these important words of the Lord: “For thus shall my church be called in the last days, even The Church of Jesus Christ of Latter-day Saints.”1

With these words, the Lord makes clear that this is not only a formal title but also the name by which His Church is to be called. Given His clear declaration, we should not refer to the Church by any other name, such as “Mormon Church” or “LDS Church.”

The term Mormon can be appropriately used in some contexts to refer to members of the Church, such as Mormon pioneers, or to institutions, such as the Mormon Tabernacle Choir. Church members are widely known as Mormons, and in interactions with those not of our faith, we may fittingly refer to ourselves as Mormons, provided we couple this with the full name of the Church.

If members learn to use the correct name of the Church in connection with the word Mormon, it will underscore that we are Christians, members of the Savior’s Church.

Brothers and sisters, let us follow up and develop the habit of always making it clear that we belong to The Church of Jesus Christ of Latter-day Saints.

Especially note this section:

we should not refer to the Church by any other name, such as "Mormon Church" or "LDS Church."

The LDS church tends to waffle a bit on whether or not the term is appropriate. Leaders of the church often teach members to not use the term. But then, at other times, the term is used by leaders of the church. Consider the recent "I'm a Mormon" campaign. If the leadership of the church found the term to be legitimately offensive, then it seems strange that they would willingly brand themselves as such in a nationwide recruitment campaign.

As a former Mormon and former full-time missionary for the LDS faith, I can state that the term is used regularly within groups of Mormons themselves. We never shied away from the term, and regularly introduced ourselves as such when serving as a missionary. There are some who avoid the term, but I would say that they are in the minority. You'll be fine using the term.