Safety before anything else
If the roof is wet, don't go up. Wait for dry conditions. A fall from a roof is a far worse problem than a leak. If the storm is still happening or you're not confident on a pitched surface, call a roofing company that offers emergency services. Many do. It's worth the cost to not end up in an ambulance.
If you do go up, wear rubber-soled shoes and have someone on the ground. Move slowly, stay away from the ridge if the pitch is steep, and don't lean over edges. The roof can look stable and still have soft spots where decking has been compromised.
What to buy
You want a heavy-duty polyethylene tarp marked at least 6 mil thickness. Get one that's large enough to cover the damaged area with at least 4 feet of overlap in every direction, and that extends over the ridge of the roof. You'll also need 2x4 lumber to anchor the tarp at the ridge, and roofing nails or cap nails for the edges.
The thin blue tarps sold in multipacks at hardware stores are not suitable for roof use. They tear in wind, don't hold fasteners reliably, and fail quickly. Look specifically for "heavy duty polyethylene" or "roof tarp" on the packaging. The price difference is small and it matters a lot in practice.
How to position the tarp
Lay the tarp over the damaged area so it extends at least 4 feet past the damage on all sides. The most important thing is that it goes over the ridge of the roof. If it doesn't, water will pool at the high edge and run back underneath. Pull it over the peak and let it hang down several feet on the other side.
If getting it over the ridge isn't possible because of the roof's geometry, you need to create a sealed edge at the high side. Take a 2x4 board, fold the tarp over it several times to create a thick rolled edge, and nail through the board into the roof deck along the ridge line. This creates a barrier water can't get under. Don't skip this step.
The tarp needs to extend past the ridge and be fastened along every edge. A tarp only secured at the corners will lift in wind and cause additional damage to surrounding shingles.
How to secure the edges
Fold each edge of the tarp under itself to create a double layer, then nail or staple through both layers into the roof deck. Space fasteners every 12 inches around the full perimeter. A tarp held only at the corners will flap in wind, the movement will abrade surrounding shingles, and it'll eventually come loose entirely.
If you're concerned about causing additional nail holes, you can weight the edges with sandbags or more 2x4 boards running lengthwise. That's less reliable than fastening but better than corners-only.
The two most common reasons emergency tarps fail: not going over the ridge (water pools at the high edge and backs under), and edges that aren't fastened (they lift in wind and expose the damaged area).
A properly installed tarp on a calm day takes about 45 minutes for most residential roof sections.
What the tarp doesn't fix
A tarp stops water from entering through the damaged area. It doesn't repair the underlying cause, address damage to flashing or pipe boots, or fix anything that was already getting wet before you put it up. Water that reached the decking or insulation before you tarped is still there. The tarp buys you time for a proper repair. It's not a solution.
How long it holds
A well-installed heavy tarp in normal conditions holds for several weeks. But check it after any significant wind event. Corners can lift, the tarp can shift on the ridge, and you won't know unless you look. If a roofer can't get to you for more than two to three weeks, check it regularly and re-secure any sections that have moved.
If you're in a situation where you're waiting longer than that, it's worth calling around more aggressively for roofers or asking about emergency repairs that could at least seal the most critical areas before a full job can be scheduled.