There are many methods to prove that a language is not regular, but what do I need to do to prove that some language is regular?

For instance, if I am given that $L$ is regular, 
how can I prove that the following $L'$ is regular, too?

$\qquad \displaystyle L' := \{w \in L: uv = w \text{ for } u \in \Sigma^* \setminus L \text{ and } v \in \Sigma^+ \}$

Can I draw a nondeterministic finite automaton to prove this?

Yes, if you can come up with any of the following:


deterministic finite automaton (DFA),
nondeterministic finite automaton (NFA),
regular expression (regexp of formal languages) or
regular grammar


for some language $L$, then $L$ is regular. There are more equivalent models, but the above are the most common.

There are also useful properties outside of the "computational" world. $L$ is also regular if 


it is finite,
you can construct it by performing certain operations on regular languages, and those operations are closed for regular languages, such as


intersection,
complement,
homomorphism,
reversal,
left- or right-quotient,
regular transduction


and more, or
using Myhill–Nerode theorem if the number of equivalence classes for $L$ is finite.


In the given example, we have some (regular) langage $L$ as basis and want to say something about a language $L'$ derived from it. Following the first approach -- construct a suitable model for $L'$ -- we can assume whichever equivalent model for $L$ we so desire; it will remain abstract, of course, since $L$ is unknown. In the second approach, we can use $L$ directly and apply closure properties to it in order to arrive at a description for $L'$.
