The number "I" means the chord from the first note of the scale (e.g. C E G in C major, or F A C in F major), and uppercase means major and lowercase means minor. Other numbers will then be e.g. "V" will be G B D in C major. You may then add digits as well in which case they indicate intervals above the bass, e.g. "V6" is a first inversion chord (e.g. B D G in C major) and "V7" adds the seventh (e.g. G B D F in C major).

You're talking about figured bass, which is its own type of notation.

"V6" to a jazz player would not indicate first inversion, it would be a major triad (built from the 5th of the tonic scale) with the addition of its own 6th scale degree. "V7" would include the dominant 7th (as opposed to the major 7th), "V13" would have the dominant 7th and also the 6th. Inversions aren't specified, the voicings are left up to the player.