CSS Positioning element relative to grandparent

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.

Basics

For controlling "cellpadding" in CSS, you can simply use padding on table cells. E.g. for 10px of "cellpadding":

td { 
    padding: 10px;
}

For "cellspacing", you can apply the border-spacing CSS property to your table. E.g. for 10px of "cellspacing":

table { 
    border-spacing: 10px;
    border-collapse: separate;
}

This property will even allow separate horizontal and vertical spacing, something you couldn't do with old-school "cellspacing".

Issues in IE ≤ 7

This will work in almost all popular browsers except for Internet Explorer up through Internet Explorer 7, where you're almost out of luck. I say "almost" because these browsers still support the border-collapse property, which merges the borders of adjoining table cells. If you're trying to eliminate cellspacing (that is, cellspacing="0") then border-collapse:collapse should have the same effect: no space between table cells. This support is buggy, though, as it does not override an existing cellspacing HTML attribute on the table element.

In short: for non-Internet Explorer 5-7 browsers, border-spacing handles you. For Internet Explorer, if your situation is just right (you want 0 cellspacing and your table doesn't have it defined already), you can use border-collapse:collapse.

table { 
    border-spacing: 0;
    border-collapse: collapse;
}

Note: For a great overview of CSS properties that one can apply to tables and for which browsers, see this fantastic Quirksmode page.