Originating in Dave Arneson's First Fantasy Campaign, Alignment is a concept used by role-playing games used to grade the personality of a character. Several models for this system exist: the most famous is the 3x3 grid used by Dungeons and Dragons from 1st edition AD&D until D&D 3.5.
This particular system grades a character in two ways: whether they are Lawful, Chaotic or in between, and if they are Good, Evil and in between. This nets for a total of nine alignments, and all beings in the setting have one of these alignments. Their names are formed by adding the two words together: a character who is Lawful but neither good nor evil is Lawful Neutral, while a character who is Chaotic and Good is Chaotic Good. The double Neutral combination is called either True Neutral or just Neutral.
Some characters are more firmly in their alignment than others; this can lead to characters being what fans have called the "Stupid" alignments: Lawful Stupid is following the law to the letter to the point of it being counterproductive in the large scheme of things, Stupid Evil is doing evil for the sake of doing evil, making the character look petty or cartoonishly evil, Chaotic Stupid is doing things that no normal person would do for the sake of "random", Stupid Good which borderline pacifism and non-violence that hinders its own goals, and Stupid Neutral which can turn on its own works in order to "preserve balance".
Alignment is a hotly debated topic, mainly because it is perceived as an oversimplification of the human nature that has been explored by all kinds of writers, scholars, philosophers, theologists and sociologists. Some players feel limited by this system and downplay its importance. Other systems do not contain such a system at all, giving the players more freedom to do what they want with their characters.
Mechanically alignments can have some degree of influence on the game: certain spells or items only affect certain alignments or cannot be used by them, clerics need to have a certain alignment in order to gain spells from their god, and some classes only get their features if they are of a certain alignment.
Please note that questions about "what alignment is person/act/society X" are off topic due to subjectivity, see What Kinds Of Alignment Questions Are On Topic? for more.