I keep talking about the HSI, and implying that it is somewhat replaceable with an "HI" or a "DG" so I think I'll try to make this all clearer. An HSI is an HI/DG combined with a VOR receiver. I promise to explain. Or at least make it muddier in an acronym-rich way.
HI stands for heading indicator, which does just what it says: indicates what heading the airplane is on, which means which direction the pointy end is pointing. The face of the instrument is a circle numbered with angular degrees around the outside of its face. Most have a tick every ten degrees and a number every thirty, except that the number is abbreviated by leaving off the final zero, so "3" is thirty degrees and "15" is 150 degrees. They fit on the instrument better that way and you learn to read it with no trouble. The numbers are written right side up at the top and upside down at the bottom, because the way the heading indicator indicates is that the disc rotates such that the current heading is constantly at the top.
DG stands for directional gyro, which is a synonym for heading indicator, "directional" referring to heading and "gyro" to gyroscope, which is what is inside the instrument, behind the disc. On start up you set the indicated heading on the DG to match the one reported by the compass. The spinning gyroscope maintains its orientation in space despite the motion of the airplane, so the difference between the position of the airplane and the initial position of the gyro is used to drive the heading card. It can get out of sync after a while, so if it isn't slaved to a fluxgate compass, you have to reset it to a more stable source of heading information, usually a magnetic compass, ever 15 minutes or so.
A VOR receiver is an instrument that indicates your angular displacement from a selected track to or from a VOR broadcasting station on the ground. You select the track with the OBS (a.k.a. "knob") and the displacement is shown by the CDI (a.k.a "needle") swinging across the instrument between zero and ten degrees in either direction, with "ten degrees" meaning "ten or more." Plus there's an additional recognizable indication if you happen to be exactly 90 degeees from the selected track, or directly overhead the station, or the instrument is broken. Same indication, on the "To/FROM indicator." I have no idea why the TO/FROM indicator doesn't have a silly name like "TFI" or "XJQ." VOR, by the way, is pronounced "vee- oh-are" and it doesn't matter what it stands for.
Note that the angular displacement is the angular displacement of the aircraft position, not the angle the aircraft heading makes with the selected track. The reading on the VOR is not affected by the aircraft heading. This can be confusing on a standard VOR because if the indication is that the aircraft is five degrees left of the track to the VOR, but the airplane is not facing towards the VOR, the pilot may need to turn left to get on track. The proper way to use the instrument is to look back and forth between the VOR and the heading indicator, do a little math and then turn to the correct heading for the intercept.
The HSI puts the VOR CDI and the OBS right on the DG. (Yay, I got all the abbreviations in the same sentence). It does the calculation for you, flipping the CDI to the correct side of the instrument for the heading, so the pilot can turn towards the needle no matter which way she is flying. So really, despite its vagueness, the name "horizontal situation indicator" accurately describes its function.