Other than basic Energy (and ☆ Pokémon), you can play up to 4 copies of a card in your deck. How many is the right number? The answer, of course is "it depends", which then begs the question "what does it depend on?" Here we're going to look at some factors which would push you towards playing 1, 2, 3, or 4 copies of a card.
If you play 4 copies of a card, it's usually a card that you want to see early, and/or see multiple times per game. This could be because the card is a crucial part of your strategy, or has a very high raw power level. Some examples:
The Basic or final Stage form of your main attacker
Holon Transceiver (for Holon Mentor) to grab 3 Basics on turn 1/2
Pow! Hand Extension
Rocket's Admin.
Rare Candy
Cessation Crystal
Coin flip Trainers, e.g. Energy Removal 2, Pokémon Reversal, Super Scoop Up
Double Rainbow Energy
2-0-4 Pidgeot. The idea is that you're usually setting Pidgeot up first, so having 4 copies maximizes the chance of drawing it naturally instead of having to spend an Elm/Celio to search for it.
If you play 2 copies of a card, it's usually a card that you want to see once per game, and can search it out with another card, but don't want to play 1 copy and risk having it Prized. Some examples:
Holon Mentor
Lanette's Net Search/Pokémon Fan Club, when you play the Castaway toolbox
Jirachi
2-2 support lines, e.g. Electrode ex
2-0-2 or 2-1-2 support lines, e.g. Pidgeot
4-2-4 or 4-2-3 lines. Usually you would just use one of your 4-of Rare Candy to jump from Basic to Stage 2, but if you get Trainer locked, you can't play Rare Candy. Having 2 Stage 1s gives you the out of searching for them with the same cards that you use to search for Stage 2s and evolving the slow way. Stage 1s are also good to have against devolution effects like Ancient Technical Machine [Rock] or Espeon ex.
Important Supporters, when you play Lapras
Important Pokémon Tools (e.g. Fluffy/Balloon Berry to switch 2 Retreat Cost starters out), when you play Castaway
Important Stadiums, when you play Scott
If you play 2 copies of a card, the chance that you Prize both of them is less than 1%. If you play 1 copy of a card, the chance that you Prize it is 10%.
If you play 1 copy of a card, it's usually a card that you can search for, or have a lot of time to draw into. Late-game cards are less susceptible to being Prized since you're going to put Prizes into your hand as you KO your opponent's Pokémon. In terms of function, 1-ofs may let you convert search effects into something else, or only be effective against certain cards/decks. Some examples:
Basic Pokémon techs that you can get with Holon Mentor: Holon's Voltorb/Magnemite, Aipom, Lapras, Lunatone + Solrock, Girafarig.
2-1-2 lines, e.g. Pidgeotto. Like 4-2-3 or 4-2-4 lines, the 1-of Stage 1 gives you an out against Trainer locks and devolution.
1-1 lines, e.g. Dugtrio. You're not going to need them in every matchup (you don't need Dugtrio if your opponent can't snipe your Bench), but they can be very impactful in those that you do end up using them in.
Evolved Pokémon techs that you play because you already have the Basics for them. E.g. Banette (Safeguard) in Banette ex for opposing ex's, Ludicolo (Happy Dance) in Ludicargo to heal off some damage from your opponent's spread damage attackers or Cursed Stone.
Pokémon Retriever
Other narrower Pokémon Tools/Stadiums, when you play Castaway/Scott. E.g. Crystal Shard, which you might need to get past Resistance, but is a blank if your opponent doesn't have anything that resists your attacker nor is weak to Colorless.
If you play 3 copies of a card, it's usually the result of trimming a 4-of because it's dead in multiples/late-game, or incrementing a 2-of because of additional synergy present in your deck. Some examples:
Great Ball, Dual Ball. You might need these to search for the Basics of your 1-1 or 2-2 lines early, but in the endgame, Balling for a Basic doesn't accomplish much if you are out of Evolutions or Energy to make use of that Basic.
Celio's Network, Professor Elm's Training Method. In decks with Pidgeot, these can be trimmed to 3 or even 2, as you just need to get Pidgeot out first, then use Quick Search instead of Supporters to grab missing Evolutions. You don't want too many Evolution-searching Supporters, since they are neutral instead of positive on card advantage unlike other Supporters.
Holon Mentor. This is usually played as a 2-of, but some decks have so many Basics, typically Holon's Castform/Magnemite/Voltorb, that it can be bumped up to 3.
3-3-3 Dark Tyranitar, 3-1-3 Machamp alongside Electrode ex. The rationale for playing 3 copies of the Stage 2 is similar to that of playing 2 of other cards, but with all numbers increased by 1: you usually need 2 of these Stage 2 attackers throughout the game (yes, really, they are that strong, the only problem is that they are Energy-hungry, which is where Electrode ex comes in), but if you play just the bare minimum of 2 copies, then you risk one getting Prized, which can hinder the setup of the second copy of the Stage 2 that you're going to throw at your opponent.
If you do end up needing 3 Stage 2s, you should have taken many Prizes or drawn into Pokémon Retriever by then, so the risk of not having the 3rd Stage 2 is low.
If you play 2 copies of a card, the chance that you Prize at least 1 (leaving only 0 or 1 copy in the hand/deck) is 19.152%. If you play 3 copies, the chance that you Prize at least 2 (again, leaving only 0 or 1 copy in the hand/deck) is 2.425%.
3-3-3 Dark Tyranitar in Bombtar, 3-3-3 Dark Dragonite in Dragtrode. These decks don't play Rare Candy, so they need more copies of the Stage 1.
3-2-3 Pidgeot in Queendom. Queendom is a Stage 2 deck, so it has Pidgeot. Nidoqueen's Power Lariat does extra damage if you have more Evolved Pokémon in play. How do you accomplish that? Simple, thicken the Pidgeot line.
Sometimes, you may end up playing a split of different versions of one particular Pokémon. Be sure to apply the guidelines above in a manner that makes sense. If you see 2 Ralts and 2 Ralts δ in a decklist, you shouldn't try to apply the rules for 2-ofs to each of them and wonder "do I really need to see each Ralts only once per game? Am I really scared of having either type of Ralts Prized?" No, you simply have 4 Ralts total, because it's the Basic form of your main attacker.
If you see a 3-1 split, it could be because the 1-of is a tech, and the other 3 is the maximum number of copies left. The aforementioned Ludicolo is a good example; you have 1 Happy Dance Ludicolo for the healing, then 3 Swing Dance Ludicolo for attacking and drawing cards.