The most important part is the concepts. Once you understand how the building blocks work, differences in syntax amount to little more than mild dialects. A layer on top of your regular expression engine's syntax is the syntax of the programming language you're using. Languages such as Perl remove most of this complication, but you'll have to keep in mind other considerations if you're using regular expressions in a C program.
If you think of regular expressions as building blocks that you can mix and match as you please, it helps you learn how to write and debug your own patterns but also how to understand patterns written by others.
Conceptually, the simplest regular expressions are literal characters. The pattern N matches the character 'N'.
Regular expressions next to each other match sequences. For example, the pattern Nick matches the sequence 'N' followed by 'i' followed by 'c' followed by 'k'.
If you've ever used grep on Unix—even if only to search for ordinary looking strings—you've already been using regular expressions! (The re in grep refers to regular expressions.)
Adding just a little complexity, you can match either 'Nick' or 'nick' with the pattern [Nn]ick. The part in square brackets is a character class, which means it matches exactly one of the enclosed characters. You can also use ranges in character classes, so [a-c] matches either 'a' or 'b' or 'c'.
The pattern . is special: rather than matching a literal dot only, it matches any character†. It's the same conceptually as the really big character class [-.?+%$A-Za-z0-9...].
Think of character classes as menus: pick just one.
Using . can save you lots of typing, and there are other shortcuts for common patterns. Say you want to match non-negative integers: one way to write that is [0-9]+. Digits are a frequent match target, so you could instead use \d+ match non-negative integers. Others are \s (whitespace) and \w (word characters: alphanumerics or underscore).
The uppercased variants are their complements, so \S matches any non-whitespace character, for example.
From there, you can repeat parts of your pattern with quantifiers. For example, the pattern ab?c matches 'abc' or 'ac' because the ? quantifier makes the subpattern it modifies optional. Other quantifiers are
Putting some of these blocks together, the pattern [Nn]*ick matches all of
The first match demonstrates an important lesson: * always succeeds! Any pattern can match zero times.
A quantifier modifies the pattern to its immediate left. You might expect 0abc+0 to match '0abc0', '0abcabc0', and so forth, but the pattern immediately to the left of the plus quantifier is c. This means 0abc+0 matches '0abc0', '0abcc0', '0abccc0', and so on.
To match one or more sequences of 'abc' with zeros on the ends, use 0(abc)+0. The parentheses denote a subpattern that can be quantified as a unit. It's also common for regular expression engines to save or "capture" the portion of the input text that matches a parenthesized group. Extracting bits this way is much more flexible and less error-prone than counting indices and substr.
Earlier, we saw one way to match either 'Nick' or 'nick'. Another is with alternation as in Nick|nick. Remember that alternation includes everything to its left and everything to its right. Use grouping parentheses to limit the scope of |, e.g., (Nick|nick).
For another example, you could equivalently write [a-c] as a|b|c, but this is likely to be suboptimal because many implementations assume alternatives will have lengths greater than 1.
Although some characters match themselves, others have special meanings. The pattern \d+ doesn't match backslash followed by lowercase D followed by a plus sign: to get that, we'd use \\d\+. A backslash removes the special meaning from the following character.
Regular expression quantifiers are greedy. This means they match as much text as they possibly can while allowing the entire pattern to match successfully.
For example, say the input is
"Hello," she said, "How are you?"
"Hello," she said, "How are you?"
You might expect ".+" to match only 'Hello,' and will then be surprised when you see that it matched from 'Hello' all the way through 'you?'.
To switch from greedy to what you might think of as cautious, add an extra ? to the quantifier. Now you understand how \((.+?)\), the example from your question works. It matches the sequence of a literal left-parenthesis, followed by one or more characters, and terminated by a right-parenthesis.
If your input is '(123) (456)', then the first capture will be '123'. Non-greedy quantifiers want to allow the rest of the pattern to start matching as soon as possible.
(As to your confusion, I don't know of any regular-expression dialect where ((.+?)) would do the same thing. I suspect something got lost in transmission somewhere along the way.)
Use the special pattern ^ to match only at the beginning of your input and $ to match only at the end. Making "bookends" with your patterns where you say, "I know what's at the front and back, but give me everything between" is a useful technique.
Say you want to match comments of the form
-- This is a comment --
-- This is a comment --
you'd write ^--\s+(.+)\s+--$.
Regular expressions are recursive, so now that you understand these basic rules, you can combine them however you like.
†: The statement above that . matches any character is a simplification for pedagogical purposes that is not strictly true. Dot matches any character except newline, "\n", but in practice you rarely expect a pattern such as .+ to cross a newline boundary. Perl regexes have a /s switch and Java Pattern.DOTALL, for example, to make . match any character at all. For languages that don't have such a feature, you can use something like [\s\S] to match "any whitespace or any non-whitespace", in other words anything.