Expository preaching is a style of preaching where the pastor explains or interprets a passage of the Bible. Usually, it is done as part of a series, so that an entire, larger section of the Bible is explained over the course of several sermons. Often, a Church will take as much as a half of a year or a year (or more) for this series of sermons. The pastor will use other Bible verses to support the interpretation of the primary passage.
A lot of people will read that and think, "so what?", so the pastor needs to preach the Bible? Isn't that what "preaching" is?
To explain, I'll present some alternative types of sermons for comparison.
The clearest alternative is called "topical" preaching. Each week, the pastor picks a different topic or picks the next sub-topic from a series. The topics use verses from around the Bible for support, so the preaching is still grounded and "Biblical". However, this reverses the preaching style. The topic determines which verses are chosen, rather than the verses choosing the topic. Also, the actual presentation is different because the purpose of the verses is to support the topic rather than be explained.
My pastor also occasionally uses a combination of the two called "expositopical" preaching where he chooses a topic or theme, chooses a primary passage, and then explains the primary passage in the same manner as the expository style of preaching.
The importance of expository preaching is that it establishes a base for the congregation to understand and interpret future sermons. By preaching through a section of the Bible in sequence and using it to determine topics will force a pastor to go over verses that they wouldn't normally use. This gives the congregation a more complete understanding of the Bible and better equips them to discuss and criticize future sermons. Remember, with topical preaching, the pastor chooses all of his verses. With expository preaching, he has to be accountable to all of the verses he has preached on before.
By actually explaining some of the passages, the the pastor also tends to explain the full meaning of the verse rather than just how it supports their argument. Sometimes, they also give alternative explanations of the verse. Those are probably some reasons why my pastor prefers his "expositopical" sermons rather than doing the normal topical style, though most of his sermons are fully expository. The expositopical sermons are saved for holidays and special dates.
Because the most important reason to use expository preaching is the foundation of Biblical knowledge in the Church, pastors can occasionally preach other types of sermons and still hold to the spirit of expository preaching.
I have visited very good Churches with topical preaching all the time. On the other hand, I have had friends who have come out of other topical Churches and are more adamant about this than I am. Its not that you cannot preach a good topical sermon. Rather, I think it will be more obvious to the congregation with expository preaching that their pastor's sermons are inconsistent or heretical and the congregation will leave or correct the situation.
I don't think that this is commanded anywhere in the Bible, so this is more like a "best practices" or "lessons learned" standard than required.