For anything over about twenty pounds that can't hit a stud, you want one of three options. Toggle bolts — the kind with the butterfly wings that spread out behind the drywall — are the workhorses. A good one-quarter-inch toggle bolt in half-inch drywall can hold fifty pounds or more. The downside is that if you ever remove it, you lose the toggle inside the wall and end up with a slightly bigger hole to patch. Snap toggles (sold under brand names like SnapSkru or Toggler) are a cleaner version of the same idea — easier to install and the toggle stays in the wall if you remove the bolt. These are what most pros reach for.
The third option, and the one worth knowing about for anything really substantial, is a molly bolt — sometimes called a hollow-wall anchor. You drill a hole, insert it, then tighten the screw until the back end collapses and grips behind the drywall. They're rated for significant weight, and they create a more stable, flush mounting point than a toggle. For a TV mount, a large mirror, or a heavy shelf bracket that can't hit a stud, a pair of mollies will often do the job cleanly.
One thing that genuinely helps when you're figuring out how to hang heavy items on drywall: pay attention to whether the load is going to be static or dynamic. A framed painting that just hangs there is static — it puts a steady downward pull on whatever's holding it. A coat rack that people will yank coats off of, or a shelf where people will grab things, creates dynamic load — repeated tugging and vibration that works fasteners loose over time. For dynamic loads, studs are worth hunting for, or you need to use more anchors than you think you do and rate them conservatively.
A few more things that'll save you headaches. Always check what's behind the wall before you drill — use an electrical stud finder with AC detection, or at minimum knock on the wall and listen for the change in sound before drilling anywhere near an outlet. Never hang anything above a bed or a couch from a single anchor point if it's genuinely heavy; two points of contact spread the load and give you a fallback if one starts to fail. And if you're in an older home with plaster walls rather than drywall, the rules change — plaster over lath needs different hardware and a gentler touch.
For most things most people want to hang — mirrors, shelving, TV mounts, large art — the answer is either studs when you can get them or quality toggle anchors when you can't. The hardware is cheap, the tools you need are minimal, and the difference between a wall that holds and one that doesn't is almost always just whether the right anchor was used for the weight. Spend five minutes on this upfront and you won't be patching drywall later.