It depends on the site. Try a right-click and then look for something like "SAVE LINK AS" or "SAVE TARGET AS" and if it offers you an audio file and you have software that will play it...you're good to go.

I have my iTunes software set up to automatically record and save a variety of podcasts to my computer so I can listen to them when I want. It's quite simple to do. You just "subscribe" (i.e., select) which podcasts to subscribe to and they are recorded and listed in your audio library. You can also sync these recordings to a player (like an iPod) so you can listen elsewhere than at your computer.

This is a special podcast which was added to a blog. The
person to whom the blog belonged, passed away two days ago
and the radio station where she was employed added the pod
cast for the benfit of her many listeners...It doesn't
have the elements you described. Excuse the terminology,
I'm a computer dummy. Anyway I listened to her for over 25 years and would like to keep this little reminder. Any
suggestions would be appreciated. Thanks

there are lots of free podcast programs(podcatchers) and many that are browser add ons.
Juice has long been one of the faves
Juice is free

List of podcatchers

What is the site? what is the name of the podcast?

here it is....http://lynnsamuels.com/

thanks

Paleblue, just right click on the podcasts and choose "Save Target As" (in Internet Explorer) to save the mp3 file.
It's a right click and 'Save Link As' in Firefox.