A regular expression is a string of characters that defines a set of one or more other strings.
Any string that is defined by a regular expression is said to match that expression.

Regular Expressions are implemented by a number of different languages and tools an unfortunately each implementation tends to be slightly different. This writeup attempts to be a general overview of REs.

A delimiter is a special character that is used to mark the beginning and end of a regular expression. A common delimiter is /.

The most basic regular expression contains no special characters other than the delimiter and matches only itself
For example
/ring/ matches ring as in spring, ringing, stringing

To get a regular expression to match more than one string you use special characters that have special meaning when part of a regular expression
The following lists special characters and their meaning together with examples. In the examples the only the strings in bold are matched. Oh and the examples build on each other

. ( period )
Matches any single character
For Example /.alk/ matches all strings with any character preceding 'alk'
as in balk or talking

* ( asterisk)
An asterisk will match zero or more occurrences of the character directly before it. Note that the character directly before it can be defined by a regular expression.
For example /ab.*c/ matches ab followed by zero or more occurrences of any character followed by c
as in abc or abjhgt gfafdg 43543 fgd c

^ ( caret )
Causes the regular expression to only match strings at the beginning of a line
For example /^T/ matches a T at the start of a line
as in
This line
but not This line

$ ( dollar sign )
Causes the regular expression to only match strings at the end of a line
For example /:$/ matches any colon that ends a line
as in
this line:
but not : this one

[] ( square brackets )
define a character class that matches any single character within the brackets.
For Example /t[aeiou].k/ matches t followed by a lower case vowel, any character and a k
as in talk or stink or teak
Within square brackets *,/ and $ lose their special meanings
If the first character following [ is a ^ it has a new meaning - the character class now matches any single character not within the brackets
Also you can use a - to denote a range of characters
For Example /[^a-zA-Z]/ matches any single character that is not a letter

Turning Special Characters Off
You can turn a special character off by preceding it with a \ ( backslash ). This is known as quoting
For Example /\*/ matches a single asterisk
and /\\/ matches a single backslash
and /and\/or/ matches and/or

Longest Match Possible
A regular expression will always match the longest match possible
For example, given the following string:
This (Dman) is a quite ( opinionated young fellow ), isn't he?

/Th.*is/ matches This (Dman) is a quite ( opiniated young fellow ), is
and /(.*)/ matches (Dman) is a quite ( opiniated young fellow )
while /([^)]*)/ matches (Dman)

NOTE YMMV: For example in Perl the Longest Match Possible doesn`t hold. Perl will match the first string it finds. Also ( is a special character in Perl's implementation so it would need to be escaped using \.