An assistant professor is a full time position. An adjunct is like the contractor/part time version of an assistant professor. The result is that adjuncts typically teach at multiple institutions, for example an R1 research university, plus a couple of local community colleges because 1 or 2 classes at one school is not enough to pay a livable wage.

In regards to credentials, in the US they can either have a PhD, be pursuing a PhD or have completed their Masters, but it depends on the institution's requirements to teach (and the classes they are teaching). You wouldn't have a Masters grad teaching graduate classes or an ABD teaching PhDs. Most well respected higher education institutions require a PhD to be an assistant professor (full time) but only be pursuing a PhD to be an adjunct/part time assistant professor teaching undergraduates.

I teach in a college math department in the US. A typical adjunct for us would have a Masters degree and teach one or two sections of Elementary Stats. So, no PhD.