How can I apply a border only inside a table?

If you are doing what I believe you are trying to do, you'll need something a little more like this:

table {
  border-collapse: collapse;
}
table td, table th {
  border: 1px solid black;
}
table tr:first-child th {
  border-top: 0;
}
table tr:last-child td {
  border-bottom: 0;
}
table tr td:first-child,
table tr th:first-child {
  border-left: 0;
}
table tr td:last-child,
table tr th:last-child {
  border-right: 0;
}

jsFiddle Demo

The problem is that you are setting a 'full border' around all the cells, which make it appear as if you have a border around the entire table.

Cheers.

EDIT: A little more info on those pseudo-classes can be found on quirksmode, and, as to be expected, you are pretty much S.O.L. in terms of IE support.

this works for me:

table {
    border-collapse: collapse;
    border-style: hidden;
}

table td, table th {
    border: 1px solid black;
}

view example ...

tested in FF 3.6 and Chromium 5.0, IE lacks support; from W3C:

Borders with the 'border-style' of 'hidden' take precedence over all other conflicting borders. Any border with this value suppresses all borders at this location.

Example of a very simple way for you to achieve the desired effect:

<table border="1" frame="void" rules="all">
    <tr>
        <td>1111</td>
        <td>2222</td>
        <td>3333</td>
    </tr>
    <tr>
        <td>4444</td>
        <td>5555</td>
        <td>6666</td>
    </tr>
</table>