Scientists measure trees for a wide variety of reasons. When I visited the Harvard Forest last week, I measured them as part of studying carbon sequestration by plants. But you can't just go out into the woods with any old tape measure and expect to collect some significant data.
That's because where you measure the tree matters. If you want to compare the diameters of two trees, you have to make sure you're measuring them in the same place. If you measured one tree at the wide base and the other further up the trunk, where trees usually get narrower, the comparison wouldn't mean much.
That's where diameter breast height (DBH) comes in. It's a way of standardizing the measuring process.
As the name implies, DBH is meant to be a diameter measurement of a tree trunk taken at, roughly, breast height on an adult. Of course, where exactly "adult breast height" is varies greatly from person to person. So DBH has been set to a standard height—1.4 meters in the United States.
In a research forest, you'll often see some kind of marker on the trees showing where this official "breast hight" is, so people can quickly move through the woods, taking diameter measurements, without having to measure vertically on each tree. In some cases, DBH is marked with yellow spray paint. In others, metal bands. These metal bands actually help measure diameter, too. Set with springs, the bands expand as the tree does, so all researchers have to is measure the distance between two dots on the band and see how far apart the dots have moved since last time.