Solid lifters are just what they say, solid, just a hunk of metal that helps convert the circular motion to vertical. A "lash" or space is needed between the rocker and valve top to account for variations caused by heat growth. These kind of cams are not designed to have pressure on the lifter when running on the "flat" or no lift part of the cycle.
A hydraulic lifter has a hydraulic cylinder in it. The top of the cylinder where the push rod sits can vary in height as needed. Each rev of the cam moves the hydraulic lifter up and down similar to a solid. The hydraulic lifter has an oil feed hole that allows oil from the engine in. This charges the cylinder. If no opposing pressure, the cylinder moves up. So why doesn't it just go to the top and stay? It is a balancing act. Oil pressure coming in, Opposing pressure from the valve spring pushing down and very slow "leaking" to handle adjustments. IE when you adjust the rocker nut down, you add extra pressure, the cylinder will "leak down". The leak is slow enough that on each rev it will not collapse, but keep pressure and adjustment for proper valve movement as needed.
When oil pressure is low, you often hear engines rattle or clack. Low oil pressure causes hydraulic lifter cylinders to collapse making a gap in the valve train and thus the clacking. Also dirty oil can cause problems with hydraulic lifters. The cylinders are precision and oil gunk can cause them to stick and not keep adjustment as they should.
So why solids vs. hydraulic today? Solids are best for high end racing applications and very high RPM. When you want 0 variance in valve timing. Most cars have hydraulic lifters or in the case of overhead cams, hydraulic adjusted followers today. Less maintenance and better overall performance and longevity on the street.
To check, take off valve cover, start engine. With the wood end of a hammer push on rocker, the push rod end. Fairly heavy push. If hydraulic this will collapse the adjuster over a few revs of the engine and you will here a lot more clacking noise. As you let go, it will pump back up over a few revs and quiet back down. If solid, gap wont change. This is also how you test if a hydraulic lifter is "stuck".
Hydraulic lifter adjustment is done with the car running at idle. Backing off till you hear the click is letting them pump all the way up. Slowly adjust to 0 lash or no gap and the clicking noise stops. Then turning down the recommended value pushes the hydraulic adjuster down to set recommended running height/load or however you want to think of it. It takes a few revs for the lifter cylinder to compensate, so turn a little slow. Turning too far bottoms out the adjuster and the valves don't close. The hydraulic adjuster allows the lifter to adjust for wear over thousands and thousands of miles. There is no "gap" with good hydraulics. The amount of the adjustment depends on the lifter manufacturer recommendations 1/4-3/4 turn. Stock style lifters take the "book" values. See instructions for your car or lifters for the adjustment turn amount. 1/4 turn is usually good if you don't have specific instructions.
Adjusting hydraulic lifters can be an oily mess if you rev the engine with the valve covers off. Idling speed is used to adjust hydraulic lifters, this will also minimize oil splash. If you want 0 oil splash, there are some clips available that can be put on the rockers to divert the oil spray when adjusting lifters. Don't forget to take them off when all done.
Solid lifters are adjusted with the engine off. The crank is turned to TDC, #1 plug fire position to start. Then 90 degree increments for each successive cylinder on a V8, based on firing order. Many solid cams have numbers for when adjusting a warm engine vs. a cold one. Also different numbers for intake vs. exhaust valves. A good mechanic that knows the cam and when the cam is on a "flat" can adjust 3 or more valves at once. Hey, he's trying to make a living. A novice should index and do 1 cylinder at a time. A little extra time is safer than making a mistake and burning valves or having the rockers too loose.
You may note non-roller lifters and push rods may "spin" a little on each lift. This is intentional, to even out wear on the bottom of the lifter.
Cams are matched to lifter type, solid, hydraulic, roller solid and roller hydraulic. They don't mix, wear and running problems can occur when they are mixed.
The type of rocker used also adds to the valve train noise while running. Stock steel stamped rockers tend to be quieter than the roller rockers.