Why five-eighths
As a vertical gets longer, more of its energy goes out at low angles towards the horizon. The best length for a vertical over a ground plane is about five-eighths of a wavelength, which gives roughly 3 dB more gain at the horizon than a quarter-wave whip. Longer than that, the current reverses near the base and the pattern breaks up into high-angle lobes. That's why 5/8 verticals are popular for 2m and 70cm mobile use, and for 10m and 6m base stations.
Matching
A 5/8-wave vertical has a reactive feed impedance, so it needs matching. The usual method is a base loading coil, often tapped, in series with a slightly shortened radiator, which also makes the antenna behave like a 3/4-wave for matching purposes. The coil depends on the radiator's thickness and height, so adjust the coil, tap point and whip length with an analyser for the lowest SWR. Many commercial 5/8 antennas use exactly this arrangement.
For a simpler antenna with no matching, use the quarter-wave vertical calculator, or for an end-fed half-wave on VHF, the J-pole calculator.
Questions
How long is a 5/8-wave vertical for 2m?
About 1.23 m for the radiator at 145 MHz, plus a base matching coil.
Does a 5/8 vertical have more gain than a quarter-wave?
Yes, about 3 dB more at low angles, which can noticeably extend VHF range.
Why does a 5/8 vertical need a coil?
Its feed impedance isn't 50 Ω and is reactive, so a base coil matches it to the coax.
Is a 5/8 vertical good on HF?
On 10m and 6m, yes. On lower bands it gets very tall, about 13 m on 20m.