Some pics will help
As you can see, there's a selector to choose the source that will be amplified and "sent" to the speakers outputs (you have two, though not selectable separately, one has lower output impedance than the other, so you can find which one matches better your speakers). The other selector, labeled "record", lets you decide which source will go through the tape out. So you might be even sending one signal through the speakers and a different one to the headphone amp or recorder.
In this one you can see all the inputs available -plenty of them- and the tape in and out. Just one tape loop. Also the speakers outputs, one with binding posts suitable for bananas or spades, and the other ones just for bananas.
For the porn fetishists, one of the insides:
Like Nate said, you can turn the volume completely down, so muting the speakers, and then listening whatever you select for the tape out, or choosing an empty input. It doesn't have a mute function as such. The tape out should work with the amp off, but it's intended to be an "always on" unit. This is one of those things which don't sound very well until they've been on for "some time"
This amp errs more to the "musical" sounding side of things than to the analytical one, but I wouldn't say it's not mainly neutral.
The DT is the model to look for. Forget the "normal" non-DT units or more modern ones.