I will annotate each statement for your information.
1.$ whereis date
date: /bin/date ...
2.$ echo $PATH
3.$ cat > date
echo "This is my own version of date."
Fri May 22 11:45:49 PDT 2009
Annotations for line:
1. "whereis date" tells us where the executable file "date" is located, according to the current $PATH value. The result tells us that the executable file "date" is found in /bin
2. echo $PATH
tells us what is the content of the $PATH environment parameter. Each path is separated by a colon ":".
3. "cat > date"
takes standard input (stdin) from the keyboard and put the keyed in content into a new file in the current directory called date. The input should be terminated by a control-d, which is not mentioned in the question.
The file "date" usually has a permission of 644 or 600 (depending on the computer implementation), which means that it is not executable.
attempts to execute the file ./date, but it is not executable (by default). Again, depending on the implementation of the system, it may return
"permission denied", or
possibly search for the next directory from the $PATH environment variable, which outputs the current date and time.