The game does have a How To Play section accessed through the main menu and a tutorial when you start the single-player mode but I thought I'd try explaining some of the card-game's basics myself here. First off, let me outline how a player's turn goes when they have one during a duel, each part known as a Phase.
Draw Phase: Draw one card from your pre-duel shuffled deck
Standby Phase: A phase usually only active if a card's effect is stated to kick in then.
Main Phase 1: Where you can bring cards into play onto your card-field and set up your actions for the next phase.
Battle Phase: Time to attack! This is where you use your in-play monsters to destroy your opponent's and knock down their Life Point total some if not completely.
Main Phase 2: What could be said as your recovery time from the Battle Phase and giving you a chance to set up defenses and counters for the next turn. Whatever you've done in the first Main Phase can usually be done here.
End Phase: This signifies the end of your turn. Some card's effects can happen here like the Standby Phase. You also have to discard any extra cards in your hand if you hold more then six at this time.
Next, let me outline the typical win conditions for a duel.
1: Drain your opponent's Life Points (or LP for short) to zero. A typical duel has both duelists starting out at 8000 LP.
2: Your opponent has run out of cards to draw at the start of their turn.
3: An automatic-win effect attached to one or more cards. Anyone who's bothered seeing the first episode of the original series has seen this in action.
Now that we got that out of the way, time for the important part of dueling: the cards!
Typically making up half of a deck's content, Monsters are the duelist's literal front-line soldiers in a duel. The two cards above show off the two basic classes of Monsters: Normal (regular monsters) and Effect (monsters which have a special ability or more). You're allowed to normally summon one monster during your turn in one of the five Monster-card spots on the field. Duelists can summon them face up as you can see above in what's called Attack Position or set them face-down and sideways in what's called Defense Position as shown below. It can also be in Defense Position right-side up, usually after it's attacked while it's face-down or if the owner changes the Position of a Monster from Attack to Defense.
Attack and Defense Positions are what bring those points by ATK and DEF on the card into play, the Position dictating which of the two numbers is active at the time. During the Battle Phase, when two Monsters collide, the active points will decide which of the two stays.
1) ATK > ATK = second Monster destroyed and it's owner takes LP-damage based on the point difference.
2) ATK = ATK = both Monsters are destroyed but no LP-damage for either Monster's owner.
3) ATK > DEF = second Monster destroyed but no LP-damage for the owner unless an effect says otherwise.
4) ATK < DEF = both Monsters survive but the attacking Monster's owner takes LP-damage based on the point difference.
5) ATK = DEF = both Monsters survive but no LP-damage for either Monster's owner.
The symbol on the right-hand corner is the Monster's Attribute: Earth, Fire, Dark, Light, Water, and Wind. The term in the brackets identifies the Monster's Type, of which there's twenty-two. Types and Attributes are used as targets for card effects and some duelists also use them to help focus what their deck is built around.
As for the 'dragonballs' you see on the two Monster Cards above, those are a Monster's Star Level. They matter when it comes to summoning them normally. A Monster with one to four stars can be dropped on the field with no problem. Those with five or six however need to be Tribute Summoned (which counts as a normal summoning), basically giving up a Monster you have on the field already to bring the 5-6er out. Seven stars and up require two Monsters instead of one. Yep, it does sound a bit slow. That's why we have Special Summons! Unlike the one-Normal-Summon-per-turn rule, a player's able to Special Summon as many Monsters as they're able to. Special Summons basically depend on a card's effect, allowing you to bring out one or more Monster out from your hand, deck or discard pile (AKA the Graveyard). In fact, besides being able to Special Summon regular Normal and Effect Monsters, there's three groups of Monsters (both Normal and Effect) who's cards are in different colors then the regular yellow and brown thanks to being Special-Summon-only Monsters. In order of appearance in the card game's history...
Fusion (Purple) - Monsters summoned from tributing (typically) two (usually) specific Monsters in your hand and/or on the field with the aid of a Fusion-effect Spell Card. The original FeSC is Polymerization, but more FeSCs popped up as the game grew to offer duelists a better chance to use Fusion Monsters (since Polymerization came out WAY back in the first set). Fusion Monster cards rest in their own separate pile on the field (AKA the Extra Deck) until summoned.
Ritual (Blue) - Monsters summoned by sacrificing one or more Monsters in your hand and/or on the field via the aid of a Ritual Spell Card. With Ritual Summoning, it's not certain Monsters being the requirement but rather the Star Level as the tribute(s) must be equal or more to the Ritual Monster's. Each Ritual Monster has their own specific RSC to summon them though there are special substitute RSCs that can be used depending on the Ritual Monster's Attribute.
Synchro (White) - Monsters summoned by two or more Monsters tributed on the field, one of which must be a special type of Monster called a Tuner. Similar to Ritual Monsters, the Star Level for the tributed monsters are taken into account but they must exactly equal the Star Level of the Synchro Monster, no more and no less.
Before Effect Monsters began to outnumber Normal Monsters in the sets, Spell Cards (formally Magic Cards) were one of the two card-groups used to keep duels from being straightforward slugging matches between Monsters. While full of wonderful and earth-shattering effects, duelists tend to keep the Spell count to about one-forth of a deck's card-count, depending mostly on how crucial they are to the deck's overall strategy. They (for the most part) can only be used during the two Main Phases and can be optionally set face-down on the back row (five available slots just like the Monsters' row). Much like Monsters, Spells tend to fall under certain categories with each having a symbol on the card to identify which Spell Card belongs to which.
Normal (no symbol on the card) - The Plain-Janes of Spells.
Continuous (infinity sign) - Spell Cards which stay on the field (within the back row) and who's effects keep going until a condition stated on the card or it's destroyed by another effect.
Equip (plus sign) - Spell Cards which stay on the field and who's effects are bonded to a monster on the field. Typically destroyed when the monster is and sometimes visa-versa.
Quick-Play (lightning bolt) - Spell Cards which can be used at anytime but must be set face-down on the duelist's turn before you can use them during the opponent's.
Field (compass) - Spell Cards that work much like Continuous Spells but have their own special spot on the field. They tend to focus on specific Types, Attributes or sets of cards. Only one Field Spell Card can be active at a time and the duelist can destroy his opponent's by laying down one of his own.
Ritual (fire) - Special Spell Cards required for summoning Ritual Monsters.
The Yang to Spells' Yin, Trap Cards make up the other one-fourth of a deck (meaning a ratio of 20/10/10 if one makes a straight 40-card deck). Traps are set face-down on the back row with the intent of being used during the opponent's turn to upset his or her's own plans. Because how game-upsetting they can be, they can only be played after the turn they're set. Much like Spells, Traps also have catagories they can be split into (though not as many).
Normal (no symbol) - The Plain-Janes of Trap Cards.
Continuous (infinity sign) - Just like Continuous Spell Cards, these stick around on the field as long as their effect(s) keep going or until they're destroyed by another effect.
Counter (curved arrow) - Trap Cards used mainly to negate a card's effect and/or destroy said card.
Knowing the cards is great but knowing which way to put them together in a deck is also important. Using a random assortment of Monsters, Spells and Traps won't get duelists very far. Focusing the deck on a win-strategy will produce much better results. This can be done by using cards that relate to a specific Type or Attribute. The game helps in this regard via the use of Archetypes which are groups of Monster cards which relate to each other under a common theme, typically indentified via their name or picture. These Archetypes tend to be supported via Spells and Traps that target cards within the Archetype much like those that target Types and Attributes. No matter which way you focus your deck, they can boil down to a few basic deck-types (with some Archetypes having a mix of traits from two of the first three deck-types below).
Beatdown - The deck uses Monsters with high ATK to quickly drain the opponent's LP.
Burn - The deck uses multiple effects to drain the opponent's LP, such as Monsters that drain extra LP upon applying damage in a battle or Spells and Traps that drain a constant amount of LP during a certain phase each turn.
Mill - The deck uses multiple effects to force the opponent to discard cards from their deck to leave them unable to draw a card at the start of their turn and thus lose.
One Turn Kill - The deck uses a specific combo of cards to win the duel in one turn by draining the opponent's max LP in one shot or using an instant-win effect. Doing so in the first turn is called a First Turn Kill.
A few last things to keep in mind when creating a deck:
A) They can only be 40 to 60 cards in the proper deck.
B) The Extra Deck's limit is 15 cards.
C) There's also a Side Deck which is used to swap cards with the proper deck during Matches (best two out of three Duels). It also has a 15 card limit.
D) You can only use three copies max of a single card in a deck. The exceptions are cards placed on the Forbidden/Limited/Semi-Limited List (updated every March and September) which means 0/1/2 copies of the card can be used in the deck depending on which part of the list the card's on.