In older versions of Minecraft there was an edge of the map called the Far Lands. However, since Beta 1.8, the Far Lands act a bit differently:
In [Beta] 1.8, instead of the Far Lands starting to generate at 12,550,820
meters from the center of the map, the playable area abruptly ends at
30,000,000 meters, and fake chunks (they are not solid, they can be
fallen through) start generating. A sure sign that one has reached the
edge of the map is that lighting no longer works past the 30,000,000th
mark. Versions between Alpha 1.2.0 (Halloween Update) and Beta 1.7.3
rendered fake chunks outside of a limit of 32,000,000 meters;
attempting to walk onto them would cause the player to die in the
Void. From the beginning of Infdev all the way to the Halloween
Update, the world abruptly ended at 32,000,000 meters, and leaving the
boundary caused you to be trapped rather than die.
To get to the 'Far Lands' you can follow the instructions in the Minecraft Wikia. However, as it says in the first sentence there is no easy way to get to the edge without using commands or external programs (Though there is a long running series that attempts to walk to the Far Lands, he is currently on episode 255). It is also prone to make your Minecraft crash so try it at your own risk.
Getting to the Far Lands without the use of an external program was a
very difficult (or at least time consuming) task, as walking to there
from the center of the map would have taken approximately 820 hours
(or 34 days). Instead, a level.dat editor could be used to teleport
the player there.
The boundary between the normal map and the Far Lands (defined by when
the map started generating the distorted terrain) occurred at X/Z of
The hard limit where chunks are overwritten is at X/Z of
±34,359,738,368, which is about 23% of the distance from the Earth to
the Sun. At X/Z of ±2,147,483,648 (crashes at 2,147,483,439), item
positions, mob pathfinding and other things using 32-bit integers will
overflow and act strangely, usually resulting in Minecraft crashing.
At X/Z of ±1.798*10308, the position of the player, represented by a
double-precision floating point number, would overflow to 'infinity',
causing a complete breakdown of arithmetic. Even at far smaller
coordinates, the limited precision would cause errors in calculations.
For example, at 1016, xPosition + 1 is equal to xPosition.
Using Single Player Commands, it is very easy to get to the Far Lands
using teleportation. After pressing the chat key (default "T") to open
up the console-like input window, using the "teleport" command (or its
abbreviation, "tp") followed by X, Y, and Z coordinates will allow the
player to go wherever they want. In Release 1.3, you can turn on
cheats to teleport. For example:
To get to where the Edge Far Lands' wall was, try "/tp 12550820 129 0". Make sure either flying is turned on or damage is turned off,
as otherwise you'll fall to your death.
To get to where the Corner Far Lands were (the walls' intersection), try "/tp 12550820 129 12550820". Again, make sure
you're protected from fall damage.
Unfortunately, there was severe lag, and slower computers used to
crash upon this teleportation. Opening a GUI helped the Far Lands
render much, much faster. You can do this by pausing (pressing Escape)
or opening the Single Player Commands prompt again. When you venture
out farther above and into the far lands, the probability increases
that a "bad chunk" will appear. A bad chunk is a chunk filled with
terribly corrupt data, and is the cause of sudden lag spikes that can
easily make Minecraft crash.
Here is a question related to the Far Lands