.badge {
  display: inline-block;
  padding: $badge-padding-y $badge-padding-x;
  font-size: $badge-font-size;
  font-weight: $badge-font-weight;
  line-height: 1;
  text-align: center;
  white-space: nowrap;
  vertical-align: baseline;
  @include border-radius($badge-border-radius);
  @include transition($badge-transition);

  @at-root a#{&} {
    @include hover-focus {
      text-decoration: none;
    }
  }

  // Empty badges collapse automatically
  &:empty {
    display: none;
  }
}

vertical-align: baseline; Check vertical-align in depth.

@at-root a#{&} @at-root means the selector definition is valid outside of .badge at the root of CSS. a#{&} = a .badge.

&:empty means .badge:empty and it selects all the empty dadges.