For almost every home application today, LED is the clear choice, but it helps to understand why when comparing against the older technologies it's replaced.
Halogen globes are cheap to buy but inefficient and run hot, contributing to higher power bills and, in downlights, a genuine heat-related fire consideration if insulation clearance isn't right.
CFLs use less energy than halogen but noticeably more than LED, take a moment to reach full brightness, and contain small amounts of mercury, meaning they need careful disposal.
LED globes use up to 80% less energy than halogen, run cool, last many times longer, and reach full brightness instantly. The main downside is a slightly higher upfront cost, which is typically recovered quickly through energy savings.
For downlights especially, an LED upgrade is usually a straightforward retrofit for a licensed electrician, and one of the more cost-effective electrical upgrades available for an existing home.