Usually: An exhaust leak, often at a gasket or a rusted joint.
But it can also be: A hole in a silencer, or on some cars a failed engine mount letting the exhaust contact the body.
Roaring or droning when you accelerate
This one usually comes from the exhaust system. Most often it is a leak at a gasket or a joint that has rusted through, letting exhaust gas escape before it gets to the muffler. The leak does not have to be big. Even a small gap will let out enough noise to make the engine sound louder than it is, especially under load.
A hole in the muffler itself does something similar. Instead of a clean, quiet flow through the baffles, the gas escapes early and you get a raw, droning sound that gets louder as you accelerate and eases off when you back off the throttle.
Less often, the noise is not a leak at all. On some cars, a failed or broken engine mount lets the engine and exhaust shift under load and knock against the body or frame. That can sound like a roar or a boom that comes and goes with acceleration in a similar way, which is why it gets mixed up with an exhaust leak.
How to tell which one you have
With the car parked and the engine idling, listen and look underneath, or have someone else do it, for a spot where the ground shimmers slightly or you can hear a hissing or popping sound right at a joint or seam in the pipe. That points to a leak. If you rev the engine in park and the whole exhaust visibly rocks or jumps more than seems normal, or you notice a clunk along with the roar, that leans toward a mount problem rather than a leak.
Is it safe to keep driving
Short term, yes, this is not something that strands you. But an exhaust leak upstream of the cabin is worth getting looked at soon rather than later, because what is escaping is not just noise. Exhaust gas contains carbon monoxide, and a leak in the wrong spot can let some of it find its way into the cabin, particularly with windows cracked or the fan pulling in outside air. That is the real reason not to let this one sit.
What fixing it usually involves
If it is a gasket or a joint, that is generally a straightforward repair, replacing the gasket or the section of pipe involved. A hole in the muffler usually means replacing that section or the whole muffler, depending on how it is holed and how the rest of the system is doing. A mount problem is a different job, replacing the mount, and it is worth confirming that is actually the cause before paying for exhaust work you did not need. Either way, it is worth having someone put it on a lift and find the actual spot before guessing at parts.