E-mail, or electronic-mail, is the computer equivelent to traditional postal mail. You write a message, address it to the recipient, and send it on its way (through your e-mail server). E-mail servers read the addressing information and route it through to the recipient's mailbox, where it waits for them to collect it (much the same as real mail, where it may go through a few sorting houses before arriving at its destination).
To read mail you either use something to read it where it is (on the server), such as a web browser or mail programs designed for remote mail boxes, or you download your mail and deal with it on your own computer. Which method you use depends on your mail server (what it supports), your own software, and your preferences. Each have their advantages and disadvantages.
Remotely stored mail can only be accessed when you're connected to the internet, and it can be slow to sort through your messages, but you can deal with your mail from anywhere (you can use different computers).
Locally stored mail will be quicker to work with, can be worked on whether or not you're connected to the internet (important if that's expensive, and useful if you've got a portable computer), but you can't work on the same messages at a different computer when you've already moved them from the server to your own computer.
The basic construction of an e-mail address is a user-account name (e.g. john) at a mail hosting address (e.g. example.com), written thus: firstname.lastname@example.org And that is how you'd address an e-mail to that person.
Addressing an e-mail is as simple as that (how you actually do so with your mail program is another issue—you'll have to read its instructions about that). Other information can be written around it, if desired, and done suitably, but it's just the address which is used by the mail servers to deliver the mail.
The address has been encapsulated with so-called “pointy brackets”, which allows the mail server to ignore the other information (the preceeding “display name” and following “comment” in parentheses), because it understands this standard way of writing addresses (which part is the address, and which parts are just information for the person reading the message).
If the “display name” contains a comma, the whole display name should be enclosed with "quote marks". e.g. "Doe, John" <email@example.com>
Commas are used to separate multiple addresses (e-mails can be sent to several addresses at once), and without any quotes the mail system would think an e-mail addressed as Doe, John@example.com should be delivered to “Doe” on its own mail system, and to “firstname.lastname@example.org”.
Your own e-mail address depends on what your mail service provider has assigned to you, perhaps as you've requested it be assigned. It'll be included in your message as where the e-mail came from, and by default, will be the address used if the recipient replies to it (they can change the address a reply will go to).
E-mail, in itself, is a quite a simple affair. Any complexity usually comes from what programs you use to deal with your mail. You'll need to learn how your software works in that regards (read its guides). I don't write specific guides for specific mail programs, there's far too many different ones, there isn't one fantastic one that everyone should use, and different versions of the same program change enough that writing guides for them is a fruitless exercise.
Beyond addressing an e-mail to the specific recipient, a mail program will need configuring for what mail servers to use. This is a once-off set-up procedure, specific to your mail service (so I can't advise you what to type in, you need that information from your mail service provider—who, for most people, is probably their ISP), and won't need changing unless you change mail services, no matter what addresses you're sending e-mails to. This is the equivelent sort of thing as knowing where your local post office is (you post all your mail through them).
There's lots of nitty–gritty information about e-mail in one of my info pages about e-mail, if you want to know more.