What is replacement theology, and why do I hear it used with a negative connotation?

To quote from a few sources (including the wikipedia article), Replacement Theology (or Supersessionism) basically states that the church is the fulfillment of Israel, and Israel as a nation no longer has a purpose, since God's chosen people are now the church.

It is a Christian interpretation of New Testament verses, viewing God's relationship with Christians as being the inheritance of the Christian Church of the promises made to the Jews (or Biblical Israelites) and Jewish Proselytes.

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The tendency is to select those promises which fit church theology (like healing, prosperity, victory) and ignore those which do not (like punishment for rebellion, keeping of feasts, sacrifices). To set the record straight: the church did not yet exist when those promises were given, and they were not given to Israel as a "type" of the church until the church should inherit them. The Old Testament promises were given to Israel, and they apply to Israel. Many of them ALSO apply to the church in a general way, and many of them apply to all nations in a general way, and many of them apply only to Israel.

The term is typically applied by Dispensational Theologians when referencing Covenant Theology.

Dispensationalists believe that God is relating to the church during the church age, which we're in right now, however, he related to the Jews during the Mosaic period, using the Mosaic law. They don't believe, however, that the church has replaced Israel, but rather that the Church is a separate entity to which God relates. They view the church age as a "parenthesis" during which time, God brings about the salvation of the Gentiles, however, this is still mainly a story between God and Israel. This theology was first systematized by John Darby, and popularized by Cyrus Scofield, via the Scofield Reference Bible, which was the first major study Bible published in the U.S., hence the theology's popularity today. The epicenter of this theology today is likely Dallas Theological Seminary.

Many Covenant theologians believe the Church replaces Israel. This is the reasoning for some of the differences we see between Covenant theology, and Dispensational theology, such as child baptism, which, in Covenant theology is the analogue to Israel's Circumcision, which is, of course viewed oddly by Dispensationalists who believe Baptism to have an entirely different meaning altogether. Many people call Covenant theology, typically in a negative light, "Replacement Theology", because they view the theology as excluding and nullifying God's promises to Israel. In this way, they view that the theology is making God out to be liar. That being said, there's often some muddiness when it comes to understanding what exactly is replaced: the Old Covenant, or the Jews themselves. I tend to believe it's the former, and not the latter, though there is seem to be some Biblical evidence to support the idea that the majority of Jews today are temporarily, or at least, partially blinded. (Romans 9-11).

The danger of "Replacement Theology", like any theology, historically speaking, has been the misinterpretation and misuse of that theology for evil means. A good example of this is the extermination of the Native Americans in North America. Many of the rationalizations behind this ethnic cleansing revolved around the misunderstood and twisted idea that, since the Church replaces Israel, and many people coming to America at that time considered themselves Christians, then the Native Americans must be our Caananites who should be completely eliminated. Obviously, this is a tragedy and a great misinterpretation of Scripture for selfish gain.

Now, Dispensationalism often comes hand-in-hand with pre-millenial eschatology, which believes in a relatively complicated timeline that includes several specific things that have been prophesied regarding end times. These prophecies often include a physical return of the Jews to the land of Israel. Many Covenant theologians, are Amillenial. They tend to take Revelation more simply, viewing Christ's return as the overall gist of the book, and many of the details allegorical. Because of this, they view the modern state of Israel as a different entity altogether than the Israelites of the Bible. This effectively removes the eschatological viewpoint from the argument on whether or whether not to support Israel.

Ultimately, whether you hear this term at all, and what light it's put into, largely depends on the theological bent of those around you. I've typically heard it more around Classical or Progressive Dispensationalists, whereas most of my Covenant Theology friends typically take it as an assumed truth, though many haven't really thought out all the implications or details of it.