How does it work?
Take a look at automata theory
In short, each regular expression has an equivalent finite automaton and can be compiled and optimized to a finite automaton. The involved algorithms can be found in many compiler books. These algorithms are used by unix programs like awk and grep. Most modern programming languages (Perl, Python, Ruby, Java (and JVM based languages), C#) do not use this approach. They use recursive backtracking approach, which compiles a regular expression into a tree or a sequence of constructs representing various sub-chunks of the regular expression. Most modern "regular expression" syntaxes offer backreferences which are outside the group of regular languages (they have no representation in finite automata), which are trivially implementable in recursive backtracking approach.
The optimization does usually yield a more efficient state machine. For example: consider aaaab|aaaac|aaaad, a normal programmer can get the simple but less efficient search implementation (comparing three strings separately) right in ten minutes; but realizing it is equivalent to aaaa[bcd], a better search can be done by searching first four 'a' then test the 5th character against [b,c,d]. The process of optimization was one of my compiler home work many years ago so I assume it is also in most modern regular expression engines.
On the other hand, state machines do have some advantage when they are accepting strings because they use more space compared to a "trivial implementation". Consider a program to un-escape quotation on SQL strings, that is: 1) starts and ends with single quotation marks; 2) single quotation marks are escaped by two consecutive single quotations. So: input ['a'''] should yield output [a']. With a state machine, the consecutive single quotation marks are handled by two states. These two states serve the purpose of remembering the input history such that each input character is processed exactly only once, as the following illustrated:
S1->*->S1, output *, * can be any other character
S2->'->S1, output '
S2->*->END, end the current string
So, in my opinion, regular expression may be slower in some trivial cases, but usually faster than a manually crafted search algorithm, given the fact that the optimization cannot be reliably done by human.
(Even in trivial cases like searching a string, a smart engine can recognize the single path in the state map and reduce that part to a simple string comparison and avoid managing states.)
A particular engine from a framework/library may be slow because the engine does a bunch of other things a programmer usually don't need. Example: the Regex class in .NET create a bunch of objects including Match, Groups and Captures.