The dot denotes that the selector is a class. So it will select elements in your page as such:
.awesome-text-box {
}
<div class="awesome-text-box"></div>
Whereas without the dot denotes an element name. Such as:
div {
}
<div></div>
In the other example you gave, the dot notation is using chaining
this is where you can select an element with numerous conditions. In your example:
p.one {
}
// Will find
<p class="one"></p>
// However it will not find
<div class="one"></div>
Whilst I am here I can give you a list of other common selectors too:
#awesome-text-box
=><div id="awesome-text-box"></div>
=> ID.btn.btn-style-1
=><span class="btn btn-style-1"></span>
=> Chaining classesp > span
=><p><span></span></p>
=> Childp span
=><p><a><span></span></a><span></span>
=> Descendant (anything below)p + span
=><p></p><span></span>
=> Sibling