First, it's important to note that not every person who gets arrested gets their picture taken. Some people are simply cited and released. In addition, arrest information is not necessarily public record (unless you are a celebrity or person of note), which can make these images even less accessible.
If you want to get a copy of your own individual arrest images, you can go to the jail where the initial booking took place and make inquiries (the process differs by county and state). Most likely, you'll need to make a formal records request to obtain this information.
If the mugshot is part of an ongoing investigation, it will most likely be completely exempt from any kind of public records request. Again, this depends on where you're located.
Most states keep an online database of people in the penal system, and that usually includes a picture and identifying information (date of crime, length of sentence, etc.). Go to your favorite search engine and type in your state, followed by the phrase "department of corrections", i.e.:
florida department of corrections
Once you get to your state's Department of Corrections page, you might have to do a little bit of searching for penal records. Each state has them listed differently; some might have a link to "Offender Search", or "Inmate Search". Continuing with our Florida example, we'll go with Inmate Population Search.
Every state's penal search form has a couple things in common. You will need at least a last name just to get started, and if you have a first name, you'll have even better results. Unless you have a tremendous amount of specific information, go with just a general search first; in other words, don't narrow your search by specific offense, or date of incarceration. Try a general search first and narrow it down as you get your results.
Federal Prison Information:
Federal inmates' mugshots are considered the property of the Federal government, and therefore are not allowed into the public domain. However, you can get information about the offender by using the Federal Bureau of Prisons Inmate Locator.
More sources for arrest records and mugshots online