What does // mean in a path?

/ is equivalent to //

From the POSIX specification (emphasis added):

3.267 Pathname

A string that is used to identify a file. In the context of POSIX.1-2008, a pathname may be limited to {PATH_MAX} bytes, including the terminating null byte. It has optional beginning <slash> characters, followed by zero or more filenames separated by <slash> characters. A pathname can optionally contain one or more trailing <slash> characters. Multiple successive <slash> characters are considered to be the same as one <slash>, except for the case of exactly two leading <slash> characters.

And:

4.12 Pathname Resolution

...
A pathname consisting of a single <slash> shall resolve to the root directory of the process. A null pathname shall not be successfully resolved. If a pathname begins with two successive <slash> characters, the first component following the leading <slash> characters may be interpreted in an implementation-defined manner, although more than two leading <slash> characters shall be treated as a single <slash> character.

For a nice discussion of exceptions (cygwin and directories) see: How does linux handle multiple consecutive path separators (/home////username///file)?

You can add multiple slashes to a directory and it will not change anything.

E.g. these three commands all do the same:
cd /home
cd /home/
cd /home//

After any of these my current working directory is is set to /home (check with pwd).

I guess that you have your prompt configured to do something smart and that you ran into unanticipated behaviour. What do you get when you echo $PS1 ?

Link to a comment on https://unix.stackexchange.com/ which may explain it.