You're asking about two very different categories of things: aliases and functions define things that act like commands; export marks a variable to be exported to child processes. Let me go through the command-like things first:
An alias (alias ll='ls -l') defines a shorthand for a command. They're intended for interactive use (they're actually disabled by default in shell scripts), and are simple but inflexible. For example, any arguments you specify after the alias simply get tacked onto the end of the command; if you wanted something like alias findservice='grep "$1" /etc/services', you can't do it, because $1 doesn't do anything useful here.
A function is like a more flexible, more powerful version of an alias. Functions can take & process arguments, contain loops, conditionals, here-documents, etc... Basically, anything you could do with a shell script can be done in a function. Note that the standard way to define a function doesn't actually use the keyword function, just parentheses after the name. For example: findservice() { grep "$1" /etc/services; }
Ok, now on to shell variables. Before I get to export, I need to talk about unexported variables. Basically, you can define a variable to have some (text) value, and then if you refer to the variable by $variablename it'll be substituted into the command. This differs from an alias or function in two ways: an alias or function can only occur as the first word in the command (e.g. ll filename will use the alias ll, but echo ll will not), and variables must be explicitly invoked with $ (echo $foo will use the variable foo, but echo foo will not). More fundamentally, aliases and functions are intended to contain executable code (commands, shell syntax, etc), while variables are intended to store non-executable data.
(BTW, you should almost always put variable references inside double-quotes -- that is, use echo "$foo" instead of just echo $foo. Without double-quotes the variable's contents get parsed in a somewhat weird way that tends to cause bugs.)
There are also some "special" shell variables, that are automatically set by the shell (e.g. $HOME), or influence how the shell behaves (e.g. $PATH controls where it looks for executable commands), or both.
An exported variable is available both in the current shell, and also passed to any subprocesses (subshells, other commands, whatever). For example, if I do LC_ALL=en_US.UTF-8, that tells my current shell use the "en_US.UTF-8" locale settings. On the other hand, if I did export LC_ALL=en_US.UTF-8 that would tell the current shell and all subprocesses and commands it executes to use that locale setting.
Note that a shell variable can be marked as exported separately from defining it, and once exported it stays exported. For example, $PATH is (as far as I know) always exported, so PATH=/foo:/bar has the same effect as export PATH=/foo:/bar (although the latter may be preferred just in case $PATH somehow wasn't already exported).
It's also possible to export a variable to a particular command without defining it in the current shell, by using the assignment as a prefix for the command. For example LC_ALL=en_US.UTF-8 sort filename will tell the sort command to use the "en_US.UTF-8" locale settings, but not apply that to the current shell (or any other commands).
