800 pixels by 800 pixels and need to convert them in Photoshop to 72 ppi for web use (at the same size: 800px by 800px).
There's no conversion to do. If they're 800 pixels by 800 pixels, that's it, they're the size you need. However, 300ppi at 800px by 800px is like a 2.3" by 2.3" picture, so are you sure the files are currently at 800pixels by 800pixels?
There's really no 72 dpi for web. This is an estimated resolution.
The web, and any computer in general, displays pixels with different monitors fitting different amounts of pixels into different amounts of space. 72dpi was an average to go by. If your monitor has a resolution (I really mean size in pixels, which isn't quite the same as resolution) of 1920 x 1050, then there are 1920 pixels across horizontally and 1050 vertically. So if your monitor's this size, and the display is 19" wider, then your screen displays about 100 pixels per inch.
Digital deviced just display the pixels. Physical size of those pixels will vary from device to device. DPI settings in files only affect the printed outcome.
To further explain this: Pixels are the dots, and DPI only comes into play when we're talking about how big to make those dots on paper, or measuring a doc in physical units of distance, like inches.
An 800pixel by 800pixel document at 300dpi settings still has the same amount of information in it as as an 800pixel by 800pixel document at 72dpi settings. The only difference is that when printed, the document at 300dpi will be printed as a square about 2.6" x 2.6". The 72dpi document will be about 11.1" x 11.1". The difference is, th pixels, which are the dots, are stuffed into a physical space according to their DPI.
Now, if we're working with documents that are set up in inches, as in an 8" by 8" doc, things change a bit. All bitmap images have pixels, or dots if you'd like to think of them that way. Every image that's not a vector has a height and width in pixels, and a user setting up and viewing their document in measurement of inches is just a feature for human beings to think of their documents in measuring units they're familiar with. This is dependent on having a DPI, whether it's one, or three hundred.
An 8" by 8" document in 300dpi is equal to a 2400pixel by 2400pixel document, because f
or every inch, there are 300 pixels in it.
An 8" by 8" document in 72dpi is equal to a 576pixel by 576pixel document, because there are 72 pixels in every inch.
If you were to set up an 8" by 8" document and draw on it, add rasterized text, etc, and the document was set up in inches, at 72dpi, then you converted it to 300dpi, now we're adding pixels by scaling the document up from 576pixels by 576pixels to 2400pixels by 2400pixels, and any non-vector images will lose some quality (on your screen mostly, but this is a whole other discussion), or at least not truly be the quality of a document originally designed at 300dpi.
But, if you're starting out by setting up the document in pixel measurements, then these wont change. At this point, the only affect changing the DPI settings will have is on the physical size of it when it's printed.