I recommend that you download the zips into a temporary folder on your desktop — that way, it's easy to find them after you've finished browsing the internet.
To download an object, click on its name next to the picture. If it doesn't seem to be working, or you end up with a page of gibberish, PC users can try right-clicking on the name and selecting 'Download to disk' (or your browser might say, 'Save target as ...'). On a Mac, click and hold until the Contextual Menu appears.
Drag and drop the downloaded .zip file on to StuffIt Expander — it will unstuff the .zip into an .iff file.

You're done!

Go here to see where to put it in the game.

You will need to use WinZip or any other similar utility to open the zip. I think that Iceows is very good — it's small, easy to use and free. It is also available in a variety of languages.
Iceows and Winzip are similar in their use — I am using screenshots from Iceows. The icons may look different on your compuer, too.

Find the downloaded zip file and double-click it. Iceows will open a window which looks just like a normal Explorer window. It shows you what is inside the zip — in this case, a tree from Persimmon Grove.
Highlight the object, then go to the File menu and select Extract.
Iceows will ask you where to save the extracted object. You can safely save it in the same temporary folder.
You're done!

Go here to see where to put it in the game.
Why Sims' objects come as zips
A zip is like the wrapping around a
parcel — it compresses an object to make
it smaller for sending over the internet.
It also helps protect an object from being corrupted while it is being transferred
through cyberspace.
When you have downloaded the zip, you need to open it to extract the object.
Once you have extracted the object's .iff file and saved it, you don't have to keep the zip — although many people do keep them as backups for their favourite objects.