The technique you're referring to doesn't actually have the button inside the textbox normally, the border and background of the button and textbox simply blend in with a wrapper div. Here is a basic example:
<div class="wrapper">
<input type="text" />
<button>GO</button>
</div>
.wrapper {
border:1px solid #000;
display:inline-block;
}
input,
button {
background-color:transparent;
border:0;
}