A blast line is an invisible boundary on a stage that a character cannot cross without getting KO'd, and causing an opponent to go beyond them are the standard means of KOing opponents outside of non-standard modes.
There are four blast lines, usually referred to as "bottom"/"lower", "left", "right", and "upper"/"ceiling" (alternatively, "north"/"south"/"east"/"west"). Super Smash Bros. Brawl and Super Smash Bros. 4 display a diagonal-pointing explosion if a character is KO'd in a corner, but this is only to improve the explosion's visibility.
Most stages have left and right blast lines which are far away from the edges of the stage's platforms, and so cannot readily be walked into. Some stages, however, have what are known as "walk-offs", where a stage's platform doesn't stop at a ledge and instead goes all the way to the left and/or right blast line, which allows players to "walk off" the stage's boundaries and be KO'd, with the exception of Balloon Fight, where walking off the side of the stage wraps you around to the other side. There is additionally a lower blast line which is triggered by falling below the bottom of the screen, which is what KO's characters that fail to recover after being knocked off the stage. Some stages have full floors and walls that prevent access to certain blast lines, but they still exist, should a character be glitched through the stage.
The upper blast line is different from the others in that characters can freely pass it without being KO'd in most situations; characters can freely jump and attack above the blast line (the cause of the Luigi ladder). Characters are only KO'd off the top if they are standing on platforms that leave the stage or they are being propelled upwards by knockback, a broken shield, a thrown Screw Attack, or the Wario Waft. These characteristics do not apply in the original Super Smash Bros.; the top blast line in said game will KO characters regardless of how they pass it.
Characters KO'd via the upper blast line are either star KO'd or screen KO'd. There are a few exceptions to this, however. In most battles in 1P Mode where the player fights a large team of opponents, the opponents are KO'd off the upper blast line as if it were any other blast line, instead of being star or screen KO'd. In Brawl, there's a glitch that causes characters who are KO'd over the upper blast line during the usage of a tether recovery to be KO'd with a simple colored blast. Additionally in Brawl, if a character is KO'd at the upper blast line while near the left or right blast line, they are KO'd as normal instead of star/screen KO'd.
The game's camera refuses to pan or zoom enough during gameplay to actually "see" the blast lines (although the lines themselves are invisible, as can be evidenced when the camera is rotated while pausing). When a character is between a blast line and the camera's view (or above the upper blast line without dying), they will appear in a bubble and slowly take damage. This is known as hoop damage.
In The Subspace Emissary's side-scrolling stages, the upper blast line is stripped of its unique properties; characters don't need to be in knockback to be KO'd by it and don't fall into the background or foreground. On the other hand, the blast lines are otherwise disabled if the character is not in hitstun; the camera will follow Player 1 and keep them away from the blast lines, while Player 2 will do a Space Jump instead of being KO'd. If a hit is momentum cancelled or otherwise interrupted, the character is still vulnerable to the blast lines for the full duration of the hitstun.
Items and projectiles that cross the blast lines are removed from play (except for, again, the upper blast line). A Smash Ball can freely cross blast lines until it leaves gameplay either by disappearing or being used.
In Super Smash Bros. 4, the blast lines have become much bigger; more specifically, most are about 2/3 of the distance of Final Destination. These allow for even lighter characters to live up to 200% on smaller stages such as Battlefield. It also, however, forces several characters to grab the ledge when recovering instead of just landing on the stage, allowing for many more edgeguarding techniques along with the addition of the edge trump mechanic. Also, certain returning stages like Corneria and Onett have smaller blast lines similar to that of their respective game. In addition, screen and star KO's no longer happen in the last few seconds of a match. They also will not always occur when a character is KO'd across the upper blast line.