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When you find the graphic that you like on a web page, right click on the image then choose 'Save Picture As'. and save it to a place where you can locate it later.
Now launch Illustrator and from Illustrator navigate to that image file to open it.
Hope this helps.
You have not told us what your operating system is.
In Windows, here is the 'Save Picture As' command that you get after right-clicking on an image (Yahoo Mail logo in this example)
And here is the logo after saving it to the hard drive:
Also can you tell me which website and which logo you are trying to save.
Try dragging the logo to your desktop. This should work in all operating system.
On a Mac, ctrl-click (hold down the "Ctrl" key and then click) on the image and then select "Save Image to Desktop". The image will be saved to the Desktop or your default downloads location.
Hope this helps. Let me know if you have any questions.