This explanation is courtesy of Ted Ede.
In active mode you talk to the FTP server, and ask for a file.  Over that connection, the server says to you, “Okay, I’m going to send you a file, and I’m going to send it you over a new connection to Port X on your machine”.  X is from 1024 to 65535.  Your client replies, “go for it”.

Now the FTP server tries to open that port, but speaking on a port that the NAT gateway would not be expecting a request on.  So, all of a sudden your NAT gateway receives a request on Port X, and it does not know which machine on the your net that the packet was intended for, so it denies the request, and you don’t get your file.

On linux, ip_masq_ftp, a kernel module, can be loaded to fix this.   Instead of blindly NATing away, it eavesdrops on the FTP protocol, and it knows in active mode that the server will reply to the client over the control connection with a port number which is contained in the *data* of the packet.  When a reply comes in on this port,  the ip_masq_ftp module tells the NAT code to route it to the correct client behind the gateway.  You still need to open the firewall to all incoming ports, but only if the source port is the FTP data port.

In PASV (passive) mode, port 21 is always initiated by the client for control and port 20 is always initiated by the client to receive data.  This makes it NATable.  Most FTP clients are set to Active mode by default and must be told to switch to PASV mode.

It’s kind of a misnomer to say the client is passive.   Basically, it instructs the server to be passive, telling it, “Hey FTP server, don’t get active on me!  When I want the file, I’ll open a connection to you and get it.”

So, you may be saying, why did they bother with this active baloney at all?  Like everything invented in unix, there’s more to it than meets the eye. Two properly implemented FTP servers can be made to work together from a third machine.  Using the FTP control port from machine A, you can tell the FTP server on machine B to download the contents of a directory on machine C.  This is basically how FTP mirror sites get updated.