Align an element to bottom with flexbox

You can use auto margins

Prior to alignment via justify-content and align-self, any positive free space is distributed to auto margins in that dimension.

So you can use one of these (or both):

p { margin-bottom: auto; } /* Push following elements to the bottom */
a { margin-top: auto; } /* Push it and following elements to the bottom */

.content {
  height: 200px;
  border: 1px solid;
  display: flex;
  flex-direction: column;
}
h1, h2 {
  margin: 0;
}
a {
  margin-top: auto;
}

<div class="content">
  <h1>heading 1</h1>
  <h2>heading 2</h2>
  <p>Some text more or less</p>
  <a href="/" class="button">Click me</a>
</div>

Alternatively, you can make the element before the a grow to fill the available space:

p { flex-grow: 1; } /* Grow to fill available space */

.content {
  height: 200px;
  border: 1px solid;
  display: flex;
  flex-direction: column;
}
h1, h2 {
  margin: 0;
}
p {
  flex-grow: 1;
}

<div class="content">
  <h1>heading 1</h1>
  <h2>heading 2</h2>
  <p>Some text more or less</p>
  <a href="/" class="button">Click me</a>
</div>

You can use display: flex to position an element to the bottom, but I do not think you want to use flex in this case, as it will affect all of your elements.

To position an element to the bottom using flex try this:

.container {
  display: flex;
}

.button {
  align-self: flex-end;
}

Your best bet is to set position: absolute to the button and set it to bottom: 0, or you can place the button outside the container and use negative transform: translateY(-100%) to bring it in the container like this:

.content {
    height: 400px;
    background: #000;
    color: #fff;
}
.button {
    transform: translateY(-100%);
    display: inline-block;
}

Check this JSFiddle