This quote has stimulated massive amounts of discussion in chess, but the fact that any great player would even make the claim shows how important tactics are.
So what is a chess tactic? It's harder to define than you'd think. The general idea is that tactics are either single moves are series of moves that win either checkmate, material, or some other advantage. That "some other advantage" is usually a positional consideration like opening up the king's defenses. I often compare strategy to the work of the generals, deploying their forces to make long term, global gains. Tactics would be like the hand-to-hand combat of the soldiers.
The simplest tactic is called "en prise," which means a piece is left to be taken for nothing in return. Such pieces are called "hanging" or free. By far, most beginner games are decided by attrition based on pieces left for free. Therefore, the first skill a chess player must learn is to find the free pieces left by the opponent and make sure no friendly pieces are en prise. This is harder than you'd think as players will often concentrate on their attacking plans or some intense action happening on one part of the board and forget about pieces away from the action.
The next most basic tactical idea, and one that is deceptively complicated is "counting." The idea is to see how many pieces defend and how many pieces attack a given piece on the board. If there are more attackers than defenders, then the attacker will often, BUT NOT ALWAYS, be able to win material by taking that piece. However, one must calculate how the trades happen at each step. The opponent is not forced to continue the trading cascade. If there are equal or greater defenders than attackers, that pieces is USUALLY safe. Again, one must look at the actual trades to determine if material is won or lost.
The next group of tactics are the simple motifs. They almost all involve attacking two things at once in some way. They include:
Forks
Discovered attacks
Skewers
Pins
The fun really begins with the final type of tactics which is the "combination." As the name implies, these sequences involve multiple motifs or trades followed by one of the simpler tactics. Examples include...
Removal of the Guard
Deflection
Decoy
Interference
There are many ways to train tactics. There are books of tactical problems at different levels, but most players now use software. And while you can purchase tactics software (CT-ART is a popular example), again most players use tactics servers online. These servers require logins, but most are free. A few examples are:
Chess.com only allows three tactics / day without a paid upgrade, but has other benefits. Mainly, there are thousands of other players available to play both live games and "turn-based" games which allow multiple days between moves. This is similar to old correspondence chess (chess by mail) which allows very long think times and presumably a player's best possible play. I consider this to be a great way to train, but you have to take advantage of the extra time to think. As an added bonus, on chess.com there is an analysis board that can be used to work out how different sequences of moves might occur in the future. This is not allowed in real games, and the ability to do this work in one's own head is a big part of what makes a player strong.
A player must practice tactics to get better. I suggest 15 minutes, 5 times a week. An hour once a week is not nearly as beneficial.
So for all my students that haven't already started the process, pick a tactics server and start training!