You'll encounter a lot of Minor Trainers in the game, who will challenge you to a duel. It's not usually smart to avoid the trainers unless your Pokémon are in bad shape. Trainer battles are the only way to get money (aside from using Pay Day in battle), and plus your Pokémon can get experience points.

If you're within sight of a minor trainer, an exclamation mark will appear over their heads, then they'll walk up to you and say something like "Wanna see my new Pokémon?" or the most basic, "Let's you and me rumble!" Sometimes, though, they'll say something completely out of the ordinary, that makes you wonder what the heck these game designers were thinking. Example: "I like shorts! They're comfy and easy to wear!"
These are exact quotes, people.

Oh yeah, and here's my personal favorite: "Why aren't you wearing shorts?"

WHAAAT??!?

Anyway, I've gotta put something on this page to make it useful, so I'm going to describe the different minor trainers you'll meet and what to expect from them.

Beauty Is obsessed with personal image. Uses different types of Pokémon depending on your area.

Biker Usually uses Poison Pokémon. Always found on a bike.

Bird Keeper uses Flying Pokémon. Found between Fuschia City and Bridge of Silence.

Bug Catcher uses Bug Pokémon. Most of them can be found in Viridian Forest, and so on, but there are a few between Cerulean City and Vermillion City.

Channeler uses Ghost Pokémon. Found in Lavender Town's Tower and Sabrina's Gym.

Cooltrainer I don't exactly get these people, but they seem to be more powerful versions of Jr. Trainers.

Cueball Fighting Pokémon.

Engineer uses Electric Pokémon. I'm pretty sure they're only found east of Vermillion City.

Fisherman uses Water-type Pokémon. Their sprite looks exactly like Hiker.

Gambler Usually use elemental Pokémon.

Gentleman Find 'em on the S.S Anne. Use semi-rare Pokémon like Nidoran.

Hiker uses Rock Pokémon. Look exactly like Fishermen.

Jr. Trainer (male and female) Just these cool-looking people. They can have any type of Pokémon imaginable, usually depending on the area you're in.

Juggler Uses Psychic Pokémon. Found in Koga's Gym.

Psychic Uses Psychic Pokémon. Find them in Sabrina's Gym.

Robber Found in the abandoned mansion on Cinnibar. Uses mainly fire Pokémon.

Rocker Uses Electric Pokémon.

Rocket Short for Team Rocket, these guys will stop at nothing to crush you. Found in Mt. Moon, Rocket Game Corner, and Silph Building.

Scientist uses Electric Pokémon. Found in various places, including Silph Building.

Super Nerd Can use any type of Pokémon, according to the area you're in.

Swimmer Uses Water Pokémon. Found swimming around in the water.

Tamer Uses rare Pokémon.