By the way you are phrasing and qualifying your question in terms of the word image, you are covering more territory than I think you believe.
The proper response to your question is it depends ...
Logos should generally be in a resolution-independent vector format, possibly EPS and preferably PDF where either of those file formats use used to represent a vector or vector and text representation of the logos. What you don't want is some low resolution GIF or JPEG raster image of a logo designed for a certain size at 72dpi on a web page.
In terms of other imagery, the response to your question is also dependent on how the "advertising publication" will be printed or whether it is strictly an on-line publication.
High or maximum quality (lower compression) JPEG is generally sufficient for most purposes with photographic raster imagery (i.e., not live text or vector graphics). Losslessly compressed (via LZW or ZIP) TIFF is always safe. In all cases, they should be tagged with their source ICC profile such that color management handles color conversions correctly. And yes, you could save a photographic image from Photoshop as a PDF file with either JPEG or ZIP compression, but for raster-only imagery, the PDF format doesn't have any particular benefit for your purpose.
If you are printing in black and white only, you should request that images be converted to grayscale by your customers. Generic conversions of color images to grayscale often are very problematic. Photoshop, for example, provides many adjustments for color to grayscale conversion to take into account the characteristics of any particular image such as the color contrasts, color values, etc. (In the days of film photography, professional and serious amateur photographers quite often used color filters when shooting black and white film in order to deal with the same color to black and white conversion issues!)
Image resolution is also an issue. Depending upon the process used for reproduction and the scaling of the image done in InDesign, your needs will vary. For typical high quality offset printing, an effective resolution of 300dpi (i.e., after scaling) is most often quite sufficient. For lower quality printing on newsprint or lower resolution laser printers, you can often get away with 150 to 200dpi effective resolution. Note that opening an image in Photoshop and applying image interpolation to increase resolution doesn't really help with an image that has too low of an effective resolution; you may end up with a pixelated mess.
What is important is the quality of the imagery itself. The lens should be properly focused. Lighting, exposure, and contrast are critical.
Having done exactly what you are going to be doing on a volunteer basis for some non-profit organizations, I wish you the very best of luck. It is amazing what amateur-hour crud will be submitted for advertisements. Just because a cellphone can create a JPEG image doesn't mean that whatever comes out of same is of either professional or even usable quality. (Also, try dealing with ads and or artwork created in PowerPoint!)
I am sure others with jump in here with their opinions and experience as well.
Wow, Dov thank you for all the info. I am planning on having the
publication done in color CMYK. I am only afraid at this point about the
quality of the images that I will be given by my customers. I Hope that
most of my customers will have good quality pictures on file that they
typically provide to advertising businesses like the one I am starting.
Thanks for the info,