Do LDS believe other churches to be the church of Satan?

You've had a few helpful answers already, but I'll add a couple connections to general Christian theology which may help you understand more.

Jacob was correct to pinpoint 1 Nephi 14:10 and JS:H 1:19 as the scriptures which get referenced when people (both Mormon and non-Mormon) make such claims. It's easy to see how these scriptures could be (mis-) understood that way.

But 1 Nephi 14:10 doesn't say "there are many churches: one of them is the Church of Jesus Christ of Latter-Day Saints, and the rest of them are churches of Satan." Instead it's saying "whatever associations and divisions among people there may seem to be on the surface, ultimately people are assembled into two camps: Christ's and Satan's." (Remember that the Biblical words that are translated as "church" all originally meant an assembly, a gathering-together, the literal meaning of con-gregation.)

The "church of the Lamb of God" is thus somewhat parallel to the idea in general Christian theology of the "church invisible" or some conceptions of "the communion of the saints." Claiming to be Christian, or having your name on the rolls of the institutional Church here on earth led by apostles and prophets, doesn't necessarily mean you're part of the "church of the Lamb of God." (cf. Matt 7:22.) On the other hand, there are people who are not members of the Church of Jesus Christ of Latter-day Saints - and indeed, even people who don't know Christ by name (think of the surprised righteous in the parable of the sheep and goats, Matt 25:37) - who are in Christ's camp.

The Lord's answer to Joseph Smith about other churches can be compared with what He says to Peter in Matthew 16:23, when though Peter was well-intentioned he was putting his conception of how things work ahead of God's plan. The creeds, even if written by well-intentioned men, are "an abomination" to God not necessarily because of their content, even when in error, but because they codify a human interpretation as having authority on par with or even above the revelations of God. The Lord uses sharp language in addressing that dangerous tendency, but He used even sharper language in dressing down Peter, and He clearly didn't mean that Peter was intrinsically evil (after all, this comes right after the blessing in v.16-19).

The source of this belief stems from a passage in 1 Nephi 14:10

And he said unto me: Behold there are save two churches only; the one is the church of the Lamb of God, and the other is the church of the devil; wherefore, whoso belongeth not to the church of the Lamb of God belongeth to that great church, which is the mother of abominations; and she is the who*e of all the earth.

The passage seems to indicate that there is a true church and that all other churches are the church of Satan. This scripture has been interpreted differently across church history. At times it has been specifically attributed to the Catholic Church, or other organizations.

Another scripture that may be the source of this belief is Joseph Smith History Verse 19.

I was answered that I must join none of them, for they were all wrong; and the Personage who addressed me said that all their creeds were an abomination in his sight; that those professors were all corrupt; that: “they draw near to me with their lips, but their hearts are far from me, they teach for doctrines the commandments of men, having a form of godliness, but they deny the power thereof.”

In this passage, Joseph Smith writes that God told him to join no churches, that none of them were true. This combined with the 1 Nephi passage lead to the conclusion that if none of the churches on the earth were the church of God, and churches that are not the church of God are the Church of Satan, then all churches that are not the Church of Jesus Christ of Latter Day Saints must be the church of Satan.

This link leads to a Mormon apologist article that shows that some leaders of the Church have said that 1 Nephi 14:10 refers to a specific church, and others have said that it refers generally to all people who oppose the teachings of Jesus Christ. The Church currently does not take an official position on the issue, but more contemporary quotations generally take the point of view that other churches are not the church of the devil.

In summary, this is not a current official teaching of the LDS church, though there are doubtless members of that church that do believe that teaching.

Another perspective on this (that has already been answered incredibly well by the people above!) comes from Gordon B. Hinckley's interview with Mike Wallace on 60 Minutes in 1996. Gordon B. Hinckley was, at the time, the president of the LDS church.

The particular quote that came to mind from the interview was the following:

Mr. Wallace: “How do you view non-Mormons?”

Response: “With love and respect. I have many non-Mormon friends. I respect them. I have the greatest of admiration for them.”

Question: “Despite the fact that they haven’t really seen the light yet?”

Reply: “Yes. To anybody who is not of this Church, I say we recognize all of the virtues and the good that you have. Bring it with you and see if we might add to it.”

Source (You can also find the full interview here if you want to watch it).

No, Mormons do not really believe that. Not like that. It sounds like those missionaries may have been either a bit zealous or oversimplified and overgeneralized the vision found in 1 Nephi 13-14:

5 And the angel said unto me: Behold the formation of a church which is most abominable above all other churches, which slayeth the saints of God, yea, and tortureth them and bindeth them down, and yoketh them with a yoke of iron, and bringeth them down into captivity.

6 And it came to pass that I beheld this great and abominable church; and I saw the devil that he was the founder of it.

It goes on to explain that "after [the Bible] hath gone forth through the hands of the great and abominable church, that there are many plain and precious things taken away from the book" (and Mormons do believe that the Bible is missing many "plain and precious" truths, hence the need for the Book of Mormon). The next chapter says:

9 And it came to pass that he said unto me: Look, and behold that great and abominable church, which is the mother of abominations, whose founder is the devil.

10 And he said unto me: Behold there are save two churches only; the one is the church of the Lamb of God, and the other is the church of the devil; wherefore, whoso belongeth not to the church of the Lamb of God belongeth to that great church, which is the mother of abominations; and she is the who*e of all the earth.

So this is probably the origin of what they said to you. However, the interpretation that Mormons think all other churches are of the devil is naïve. This passage only talks about two churches, not many. Further, the LDS Church acknowledges, appreciates, and cooperates with other churches that preach of God and Jesus Christ.

What the angel is telling Nephi here is that, when the Book of Mormon was coming forth and the Lord's true Church was beginning to be restored to the Earth, that there would be those who oppose it and those who join it. We have no reason to believe the vision can be interpreted this way after the beginning of the restoration of the Church, say, today's day and age. In fact, the vision Nephi had says that the numbers of the Church of the Lamb of God were few: but these days, there are over 15 million members of the LDS Church. Does that count as a "few"? Maybe, compared to the population of the world. But there's no real official interpretation of this vision. It is left for the reader to interpret as led by the Spirit.