I haven't taken any semiconductor classes, but if you're interested in an answer constrained to circuit-level operation, the quick answer is:
with NMOS, current flows from Drain-to-source (arrow points away from device at the Source)
with PMOS, current flows from Source-to-drain (arrow points to the device at the Source)
In the diagram above, the words P-channel refer to the type of channel that forms underneath the Gate. The P signifies that the channel forms on P-type semiconductor, while the N signifies an N-type semiconductor.
With respect to the confusion. you're right, it is confusing. What you're seeing is known as a source-body tied terminal. In some applications this is useful (see below for more.) Ignore it for the time being.
Generally, when examining an analog circuit schematic, it is conventional to see arrows on the Source terminal of the transistor.
When examining digital transistor-level schematics (as opposed to gate-level, i.e. AND, OR, XOR gates), conventionally, there are no arrows. The distinguishing aspect is that the PMOS will have a little bubble at the Gate terminal, while the NMOS won't have any bubble. Be assured, they are in fact the same transistors (both PMOS and NMOS) in both analog and digital applications. But the way they are operated is very different.
Fun Fact for a beginner
The transistor is a four-terminal device: Gate, Drain, Source, and Body. As an introduction to microelectronics, it is conventional to ignore the body terminal intially, but only to assist in familiarizing you with the main equations. However there is a semiconductor phenomenon known as the body-effect which introduces an added layer of complexity to hand calculations with respect to calculating the quiescent operating point of a transistor (quiescent operating point is an important word you'll encounter; it's just a fancy word that signifies the IV or current-voltage operating point of the transistor in question.)
Modeling a transistor is highly complex undertaking and is an electrical engineering or applied physics discipline in itself. Any introductory textbook in microelectronics usually starts a chapter mentioning p-n junctions (a type of doped silicon semiconductor).
If you're really interested, and have a basic grasp of quadratic equations and algebra, you may want to take a look at a great introductory textbook written by Behzad Razavi. I wish I had this book when I took microelectronics in university. However it assumes an understanding of Basic Circuits (i.e. resistors, capacitors, and inductors.)