What does "solo" and "tutti" mean in Mozart's Horn Concertos?

The two words have certain meanings, but they aren't necessarily literal in the context of a musical score. "Tutti" does mean "all", but by that time in music it had taken on the sense of "anyone or everyone apart from the soloist". In the same way, the soloist isn't literally playing solo when the orchestra is accompanying, but it's still describe like that.

If you're reading the published solo part for a concerto, the orchestral melody is often included in that part when the soloist isn't playing. In that case, the sections are indicated by "solo" when the soloist plays, regardless of the orchestra accompanying. When the soloist rests and the orchestra or part thereof is playing, that section in the part will be indicated by "tutti".

Solo means only 1 instrument is playing out of the entire group of, for example, cellos (but in orchestral pieces, there can be 2 or more solo instruments playing at the same time)

Tutti means all instruments of the entire group are playing (especially after a passage of solo playing)

1 person alone vs all of the horns together

Solo and tutti always mean the same thing - "on yer own" and "all together."

1. "Hey You!"

2. "The buddy system ... with everyone else in the section."