Old Text Book Current Question
I'm not sure if this is a formal definition, but this works pretty well in practice:


If elements have the same current running through them (they share a voltage drop) they are in series.
If elements have the same voltage across them (they share current) they are in parallel.


Edit to answer newly updated question:

Yes, you can combine series and parallel elements:







simulate this circuit &ndash; Schematic created using CircuitLab

This is an example of both a series and a parallel connection in the same circuit. In this example,
$$\text{Voltage D} - \text{Voltage F} = 10V\;\text{(due to the battery)}$$
and Voltage E is somewhere in between the two (the voltage is shared by Lamp 1 and the combination of Lamps 2&amp;3).

You can also see that the Current A is being split by the elements Lamp 2 and Lamp 3, into Current B and Current C. Since we can't create or destroy current, they add up to Current A:
$$\text{Current A} = \text{Current B}+\text{Current C}$$

So you can say that Lamp1 and the combination of Lamps 2&amp;3 are in series (Lamp 1 and the combination of Lamps 2&amp;3 have Current A in them), while the sub-circuit of Lamp 2 and Lamp 3 has a common voltage across those elements (\$\text{Voltage E}-\text{Voltage F}\$) so they are in parallel!

I've gone a little overboard on answering this question in a very general and mathematical way, but that's how people who do circuit design work with these numbers :) If anything is unclear, feel free to ask below.