On a hit and roll you remove a stone and roll on yourself: contact point, roll direction, back line and free guard zone rules, step by step.
The honest answer: a hit and roll is a takeout in which your stone knocks an opposing stone out of play and then rolls itself to another spot in play [1]. The "roll" is the sideways movement of a stone after it has struck a stationary stone [1]. So your stone not only clears, it also positions itself. A hit and roll can even be played off one of your own stones [2].
For comparison, the hit and stay: here your stone stays at the contact point (a "nose hit") [3]. The hit and roll is the variant for situations in which you do not want to stay at the contact point, for example because you want to roll behind a guard.
How hard is it? It is the takeout with an extra task, and the roll target is not 100 percent controllable. Even good teams miss it more often than they would like. Plan realistically: first the clean hit, then the direction. There is no time promise.
Play a hit and stay on a target stone: contact in the middle, your stone stays [3]. You need to know your stone reliably hits and removes the opponent before you start rolling.
Milestone: three out of five stones clear the target stone and stay at the contact point.
Decide before the throw where you want to roll: behind a guard, to the edge of the house, or away from a dangerous spot. The roll target is a specific spot your stone should reach [2].
Milestone: before every throw you say aloud where the stone should roll.
If you hit the target stone in the middle, your stone usually stays. The thinner you hit it (at the edge), the more sideways motion your stone keeps. A rule of thumb from the physics of an oblique impact, not a federation text: if you hit the target stone on its left side, you push it to the right and your own stone rolls to the left. The rotation of your stone can shift this slightly.
Milestone: out of ten throws with left contact, your stone rolls left at least six times.
With too much speed your stone rolls far, perhaps over the back line [1]. With too little, the target stays in play or your stone stays at the contact. Choose the speed at which the target goes reliably but your stone still has energy left. That is a feel only repetition brings.
The roll belongs to you: your team may keep sweeping your stone while it is in front of the tee line at the playing end [1]. Agree: who sweeps, when does he stop, and take care not to touch the stone ("burned" [1]).
Milestone: with sweepers, your stone lands in the target area in six out of ten tries.
Before every hit and roll, ask: which opposing stones are my friends? The US federation's instructor manual recommends sorting each stone as friend, enemy or neutral. Opposing stones behind the tee line can be your friend, especially if you have the hammer [4]. Use them as a backdrop for the roll.
A takeout in which your stone knocks an opposing stone out of play and then rolls itself to another spot in play [1]. It can also be played off one of your own stones [2]. The goal is often a spot behind a guard.
On a hit and stay your stone stays at the contact point [3], on a hit and roll it rolls on to another position [1]. Hit and stay is the easier shot, hit and roll additionally needs a plan, contact point and speed.
The skip directs play and is in charge of the house when the team delivers [1]. They tell you line and target. "Hurry" means sweep harder [1]. On a hit and roll they also tell you where to roll.
Set-up in the Hack in Curling: How to Start Stable