Thanks for the question!

Files are transferred/loaded to a web server via FTP (File Transfer Protocol). It used to be that you had to have a program installed on your computer to communicate with your server. I've used Cute FTP for years now and have been very happy with performance. There are a bunch of free ftp programs available on the web but free downloads are notorious for containing spyware, adware etc. so I'd stay away from them.

Internet Explorer users can use Web Dav or Windows Explorer to transfer files but I didn't know about Firefox until you asked the question. 

It turns out that there is a free extension for Firefox, FireFTP, that does exactly what you're looking for and is more convenient than the options on IE because you can drag and drop files as opposed to just being able to copy and paste.

1. Go to https://addons.mozilla.org/en-US/firefox/addon/684 and select "Install Now".

2. Restart Firefox and go to Tools > Fire FTP

3. This will open the file transfer window and you'll have to create an account for the site. Hit the drop down next to Manage Accounts and select "New".

4. This will open a screen for you to enter the details of the FTP site that were given to you by your host, or the person that manages your server. Enter the details accordingly and click "OK".

5. You will be taken back to the file transfer screen but now the account that you just created will be an option in the drop down next to manage accounts. Just select it from the drop down and click "Connect".

6. That's it, you will now see a the transfer screen with your local files in the left pane and the files on the server in the right pane. Just use the arrows, or drag and drop files to upload or download.

Use the "Browse Button" in the left pane to navigate to different directories on your computer to locate the files that you need.