I once made a device that had buttons labeled with the harmony numbers, and you could play it without knowing the literal chord equivalents of the harmony numbers in any particular key, in other words you didn’t have to know that in D, say, the 5 chord is A major. If you are a musician and you don’t have such a device, you always work with actual keys and literal chord names. Appendix 1 is a good reference for translating harmony numbers to literal chords.
And if you work with literal chord names, it helps greatly to have a good feel for the sequence of jumps by a fifth : F – C – G – D – A etc. This string of fifths can be depicted as a circle, the circle of keys (also called the 'circle of fifths'). As you go counterclockwise in the circle you are jumping to the dominant of each key (1 -> 5). If you go clockwise, you are doing 5 -> 1 transitions (or equivalently, 1 -> 4 transitions). The diagram also shows the key signature (number of sharps or flats) for each key.
People who are musically sensitive agree, for the most part, that the key of C and the keys on the left side of the circle, the 'sharp' keys, sound bright and energetic, while the keys on the right side, the 'flat' keys, have a more solid and grounded quality.
The inner circle is the set of twelve minor keys, each one aligned with its relative major.
Sometimes this diagram is shown in the opposite sense: the dominant jumps are clockwise instead of counterclockwise. I believe the sense shown here corresponds better with the intuitive notion that the dominant precedes the tonic, and 1 -> 4 is a motion forward.
