For HTML 4, the answer is technically:

ID and NAME tokens must begin with a letter ([A-Za-z]) and may be followed by any number of letters, digits ([0-9]), hyphens ("-"), underscores ("_"), colons (":"), and periods (".").

HTML 5 is even more permissive, saying only that an id must contain at least one character and may not contain any space characters.

The id attribute is case sensitive in XHTML.

As a purely practical matter, you may want to avoid certain characters. Periods, colons and '#' have special meaning in CSS selectors, so you will have to escape those characters using a backslash in CSS or a double backslash in a selector string passed to jQuery. Think about how often you will have to escape a character in your stylesheets or code before you go crazy with periods and colons in ids.

For example, the HTML declaration <div id="first.name"></div> is valid. You can select that element in CSS as #first\.name and in jQuery like so: $('#first\\.name'). But if you forget the backslash, $('#first.name'), you will have a perfectly valid selector looking for an element with id first and also having class name. This is a bug that is easy to overlook. You might be happier in the long run choosing the id first-name (a hyphen rather than a period), instead.

You can simplify your development tasks by strictly sticking to a naming convention. For example, if you limit yourself entirely to lower-case characters and always separate words with either hyphens or underscores (but not both, pick one and never use the other), then you have an easy-to-remember pattern. You will never wonder "was it firstName or FirstName?" because you will always know that you should type first_name. Prefer camel case? Then limit yourself to that, no hyphens or underscores, and always, consistently use either upper-case or lower-case for the first character, don't mix them.

A now very obscure problem was that at least one browser, Netscape 6, incorrectly treated id attribute values as case-sensitive. That meant that if you had typed id="firstName" in your HTML (lower-case 'f') and #FirstName { color: red } in your CSS (upper-case 'F'), that buggy browser would have failed to set the element's color to red. At the time of this edit, April 2015, I hope you aren't being asked to support Netscape 6. Consider this a historical footnote.

You can apply this CSS to the inner <div>:

#inner {
  width: 50%;
  margin: 0 auto;
}

Of course, you don't have to set the width to 50%. Any width less than the containing <div> will work. The margin: 0 auto is what does the actual centering.

If you are targeting Internet Explorer 8 (and later), it might be better to have this instead:

#inner {
  display: table;
  margin: 0 auto;
}

It will make the inner element center horizontally and it works without setting a specific width.

Working example here:

#inner {
  display: table;
  margin: 0 auto;
  border: 1px solid black;
}

#outer {
  border: 1px solid red;
  width:100%
}

<div id="outer">
  <div id="inner">Foo foo</div>
</div>

EDIT

With flexbox it is very easy to style the div horizontally and vertically centered.

#inner {  
  border: 1px solid black;
}

#outer {
  border: 1px solid red;
  width:100%;
  display: flex;
  justify-content: center;
}

<div id="outer">
  <div id="inner">Foo foo</div>
</div>

To align the div vertically centered, use the property align-items: center.