Twitter Bootstrap 4
In Twitter Bootstrap 4, inputs and buttons can be aligned using the input-group-prepend and input-group-append classes (see https://getbootstrap.com/docs/4.0/components/input-group/#button-addons)
Group button on the left side (prepend)
<div class="input-group mb-3">
<div class="input-group-prepend">
<button class="btn btn-outline-secondary" type="button">Button</button>
</div>
<input type="text" class="form-control">
</div>
Group button on the right side (append)
<div class="input-group mb-3">
<input type="text" class="form-control">
<div class="input-group-append">
<button class="btn btn-outline-secondary" type="button">Button</button>
</div>
</div>
Twitter Bootstrap 3
As shown in the answer by @abimelex, inputs and buttons can be aligned by using the .input-group classes (see http://getbootstrap.com/components/#input-groups-buttons)
Group button on the left side
<div class="input-group">
<span class="input-group-btn">
<button class="btn btn-default" type="button">Go!</button>
</span>
<input type="text" class="form-control">
</div>
Group button on the right side
<div class="input-group">
<input type="text" class="form-control">
<span class="input-group-btn">
<button class="btn btn-default" type="button">Go!</button>
</span>
</div>
This solution has been added to keep my answer up to date, but please stick your up-vote on the answer provided by @abimelex.
Twitter Bootstrap 2
Bootstrap offers an .input-append class, which works as a wrapper element and corrects this for you:
<div class="input-append">
<input name="search" id="search"/>
<button class="btn">button</button>
</div>
As pointed out by @OleksiyKhilkevich in his answer, there is a second way to align input and button by using the .form-horizontal class:
<div class="form-horizontal">
<input name="search" id="search"/>
<button class="btn">button</button>
</div>
The Differences
The difference between these two classes is that .input-append will place the button up against the input element (so they look like they are attached), where .form-horizontal will place a space between them.
— Note —
To allow the input and button elements to be next to each other without spacing, the font-size has been set to 0 in the .input-append class (this removes the white spacing between the inline-block elements). This may have an adverse effect on font-sizes in the input element if you want to override the defaults using em or % measurements.