Let's start out with Heading tags.  These tags are good for creating titles or section headings.  Here are some examples:

<H1>Large Heading!</H1>  will give us:

Large Heading!

Heading 2

Heading 3

Getting Small

Smaller Still...

You must have good vision...

This will generate the following:

End this line now!!
Thanks!

Basically, a line break is like hitting the "enter" key when you are writing text. The browser will not go to the next line until it runs out of space, or sees a tag that will force it to the next line. So typing the following in your text editor will display only one line of text in the browser:

Hello,
I want
a new line.

This gives us: Hello, I want a new line.

To make the text move to the next line, use your <BR> tag from above:

Hello,<BR>
I want<BR>
a new line.

Now this will do what we wanted it to:

Hello,
I want
a new line.

Now, let's move on to the paragraph tag, <P>.  This tag will skip a vertical space after going to the next line, as though you had typed <BR> twice.  This tag is good for skipping a line quickly and for knowing where you wanted a new paragraph to begin.  Here's an example:

This is some cool stuff.
< BR>
This is the next line.
< P>
This is a new paragraph.  Is this cool or what?

This will give us the following:
This is some cool stuff.
This is the next line.

This is a new paragraph.  Is this cool or what?

The paragraph tag does not require a closing tag, but if you'd like to add one for your own reference, you place a </P> where you would like the paragraph to end, like this:

<P>
This paragraph needs a visual ending!
< /P>
< P>
Here is a new paragraph....<BR>
and the end.
< /P>

This will give you the same thing as using just the opening <P> tags, like this:

<P>
This paragraph needs a visual ending!
< P>
Here is a new paragraph....<BR>
and the end.
 
Both of these will give you this:

This paragraph needs a visual ending!

Here is a new paragraph....
and the end.