DIAGNOSIS A slight wobble is not normal and means the chuck isn't gripping the bit perfectly straight. It's usually not a broken drill, but a worn or dirty chuck. Grit or a damaged jaw inside the chuck can cause this.
FIX Start by cleaning it. Open the chuck jaws fully. Use a can of compressed air or a small brush to get any sawdust or debris out from between the jaws. Put a known-straight bit (like a new drill bit) in, tighten the chuck by hand as hard as you can, then give it a test spin. Often, clearing the junk fixes it.
IF THAT DIDN'T WORK If it still wobbles with a clean, straight bit, the chuck itself is likely worn out. The jaws can lose their precision grip over time. Replacement chucks are available, and swapping one requires a hex key to remove the screw inside the chuck, then using a big wrench to unscrew the old chuck (left-hand threads, so turn clockwise to loosen).
HEADS UP A wobbly chuck will give you inaccurate, oversized holes and can wear out bits faster. It's worth fixing. Before you decide the chuck is bad, double-check that your drill bit isn't bent—that's the most common culprit.