Tires have more influence on how a bike feels than almost any other single component, and they're also one of the easiest things to change if your current setup doesn't match how you ride.
Tread pattern
Smooth or lightly textured tires roll faster and more efficiently on pavement but offer little grip off-road. Aggressive, knobby tread digs into loose dirt, mud, and gravel for traction but creates more rolling resistance and noise on pavement. Somewhere in between, semi-slick tires with a smooth center and knobbier edges are a common compromise for mixed-surface riding.
Width
Wider tires generally offer more comfort and traction, since they run at lower pressure and have a larger contact patch, while narrower tires roll faster on smooth surfaces and weigh less. What counts as "wide" varies a lot by bike category — a wide road tire and a narrow mountain bike tire can be similar widths but serve completely different purposes.
Tire pressure
Running a tire at the right pressure for its width and your weight matters as much as the tire choice itself. Too high, and the ride feels harsh with less grip on rough or loose surfaces; too low, and you risk pinch flats and sluggish rolling resistance. Most tires list a recommended pressure range on the sidewall as a starting point.
Matching tires to your riding
Be honest about the surfaces you actually ride most, not the ones you ride occasionally — a tire optimized for rare terrain at the expense of your everyday riding is usually the wrong tradeoff.