If I wanted to print an image found online and get a typical 27" x 40" sized poster what resolution image do I need? 4000 x 6000 or double that? I am confused over DPI and PPI.
It depends on how close you expect the viewer to stand, as vision is based on angular frequency. If your image has a certain size and you need to print it at some size then there is not much you can do about the resolution.
150 PPI is usually quite acceptable for items you view at a distance. Most human sized outdoor commercials are at that kind of resolution. Especially if your text items are vectors.
You could use online calculators to verify your values, if you have no idea what your doing.
Sounds like you're confusing dimensions with resolution. Go for the dimensions as close to 27x40 as possible (using the online calculator that joojaa mentioned to convert inches --> pixels). When choosing resolution for printing, 300 is best for high-res graphics or fine details, but if it's okay to lose some detail you can go with 150.