The terms Johnny, Timmy, and Spike refer to the three player Archetypes that were outlined by Magic Card Designer (among other things) Mark Rosewater in a blog post, and later revised on the Magic the Gathering website. They are used to categorize players based on what their motivations to playing the game are, what types of cards they enjoy, and what encourages them to keep playing.
The archetypes are defined as follows:
Johnny is often described as the combo player; he wants to carefully craft a plan and then execute it. Johnny players tend to gravitate towards niche cards with unique effects. Sometimes Johnnies will center decks around cards that others consider unplayable just to prove that it can be done. Johnnies are looking for a way to express themselves and their creativity, even if they don't win every time.
The common bond to all the Johnnies is that they are on a mission to show the world something about themselves. What they're showing varies tremendously, but at the core of each Johnny is a similar motivation: “Look at me, world! Look at me!” - Mark Rosewater
Timmy is described as the power player. If he wins, he wants to win big just barely scraping by doesn't interest him. Timmy players tend to gravitate to big cards with big effects: anything that will help them dominate their opponents. Though he wants to win big, he doesn't really care about having a finely-tuned and efficient deck; he just wants to have fun. If Timmy only wins a few of his games, but in those games he absolutely destroys his opponent, he can go home satisfied.
Timmy plays with cards that make him happy; cards that create cool moments; cards that make him laugh; cards that allow him to hang with his friends; cards that cause him to have fun. Winning and losing isn't even really the point (although winning is fun – Timmy gets that). For Timmy, the entire reason to play is having a good time. - Mark Rosewater
Spike is often described as the tournament player. Spike is in it to win it, and will play the best deck he can find. Spike players often look for any opportunity they can to improve their chances of winning and are considered the most competitive archetype. They are attracted to the most efficient cards and decks. They will often research what decks are currently powerful or performing well and either play them as is or analyze and tune them until they are perfect. Spikes want to win and to prove how good a player they are.
Spike plays to win. Spike enjoys winning. To accomplish this, Spike will play whatever the best deck is. Spike will copy decks off the Internet. Spike will borrow other players’ decks. To Spike, the thrill of Magic is the Adrenalin rush of competition. Spike enjoys the stimulation of outplaying the opponent and the glory of victory. - Mark Rosewater
One thing to keep in mind is that players do not necessarily fit cleanly into one category. Often times you will have mixes, such as Johnny/Spike players who are innovating new decks to be seen on the pro-tours, or more casual Timmy/Johnny players who try to brew convoluted commander decks to play with friends. The Timmy/Johnny/Spike trio is not the end-all solution for categorizing cards and players, but it is useful shorthand for sorting or describing things.
MTG Official Blog- Mark Rosewater, TIMMY, JOHNNY, AND SPIKE
MTG Official Blog- Mark Rosewater, TIMMY, JOHNNY, AND SPIKE REVISITED
GDC Talk- Mark Rosewater, Magic: the Gathering: Twenty Years, Twenty Lessons Learned