How to append one file to another in Linux from the shell?
cat file2 >> file1
The >> operator appends the output to the named file or creates the named file if it does not exist.
cat file1 file2 > file3
This concatenates two or more files to one. You can have as many source files as you need. For example,
cat *.txt >> newfile.txt
Update 20130902
In the comments eumiro suggests "don't try cat file1 file2 > file1." The reason this might not result in the expected outcome is that the file receiving the redirect is prepared before the command to the left of the > is executed. In this case, first file1 is truncated to zero length and opened for output, then the cat command attempts to concatenate the now zero-length file plus the contents of file2 into file1. The result is that the original contents of file1 are lost and in its place is a copy of file2 which probably isn't what was expected.
Update 20160919
In the comments tpartee suggests linking to backing information/sources. For an authoritative reference, I direct the kind reader to the sh man page at linuxcommand.org which states:
Before a command is executed, its input and output may be redirected using a special notation interpreted by the shell.
While that does tell the reader what they need to know it is easy to miss if you aren't looking for it and parsing the statement word by word. The most important word here being 'before'. The redirection is completed (or fails) before the command is executed.
In the example case of cat file1 file2 > file1 the shell performs the redirection first so that the I/O handles are in place in the environment in which the command will be executed before it is executed.
A friendlier version in which the redirection precedence is covered at length can be found at Ian Allen's web site in the form of Linux courseware. His I/O Redirection Notes page has much to say on the topic, including the observation that redirection works even without a command. Passing this to the shell:
$ >out...creates an empty file named out. The shell first sets up the I/O redirection, then looks for a command, finds none, and completes the operation.