Remote linking is when you do not load an image or file  to your file manager and place the image on your page. It is taking the source info, and placing it in your code.
So> rather than placing the pic in your files, You use mine. Bad thing!
That means the code looks like 
<img src="http://www.mydomain.com/stolenimage.gif">   
rather than <img src="stolenpic.gif">

Local linking is having the images used on the pages in the same server/ file system as the html pages. That means, all the files associated for the page are together. It means you right click and select save as> and upload the image to your own server or file manager. 

Now this is not a lecture about respecting the rights of the artist.
It could be, but there are hundreds of sites that boast free graphics, and they stole them. The best You can do is credit where you get images~ and get permission if you can, now... enough of that. And back to why remote linking is bad.

Now if you use a free host like geo, you risk killing your site if you "allow" remote linking. So don't even try having a download. Some well meaning person will remote link, the link gets around. And poof.. suddenly your site is gone all the files are gone, and you can no longer upload. and they don't restore anything! 
Even a small image file like a button to your page can cause a lot of problems. Encourage people to take the graphics and link back!

Why should you care?

When you remote link an image, you are basically giving the person you are taking the image from  CONTROL OVER YOUR SITE! 

That person  (like ME) can do anything to the image. move it. delete it. change its size, shape, content, basically turn your site from pretty to trash. Or worse.  Consider yourself warned.  Most webmasters will ask once, and then they will replace the pic. Often with one you wont like. 

SO> you say you do not have anyplace to host the images yourself? You have 3 choices, find a new host, ask the person who has the image if you can remote link, or finally, look deep inside yourself, and realize, maybe you don't need that pic~ you don't need what someone else has or has done. 

Realize also that if you are remote linking, a webmaster can contact your home page provider, and file a complaint. You risk losing your site if you remote link the wrong image. Just right click it and save it. and upload  to your own file system~!

you say.. no one will ever know if you remote link an image~ 

But they do! People who own  domains check their server log files
They run the statistics> for personal reasons, to credit those who links, to see who's linking, or to check files/pages popularity.

The site owner sees all the remote links! They see if you are stealing the pics, which also steals their bandwidth. !!!!! Pointing to their files is one sure way to slow down their site and yours. And bandwidth costs money. 

geo, angelfire, tripod have stopped allowing  their pages to be used for  auction pics, sig tags etc. you can use your isp space or...

free image hosting sites

• boomspeed.com

• liquid2k.com

that is why the smiles pages here have large multiple smiley pics, rather than individual ones,  because dozens of message boards allowed their users to remote link the images. 

most popular stolen image from here.