The term K-style gutters is derived from the profile of this type of guttering when viewed from the side. The outer shape of the gutter is supposed to (very vaguely) represent the letter K, as opposed to a perfectly-rounded, half-circle-shaped gutter. Like other types of gutters, K-style gutters are available in different sizes. The two most common sizes, as measured by gutter diameter, are five-inch and six-inch; though seven-inch, eight-inch, and custom sizes are obtainable from certain places. K-gutters are most often made with aluminum.
Half-round gutters look like you took a fully-enclosed pipe and cut it in half lengthwise. A "true" half-round gutter section has curled-top outer edges on both sides, although you can purchase these types of gutters where the outer edge has a curled top on only one side. Like traditional "K-style" gutters, half-round gutters are available in different widths and colors, and can be purchased in sections or installed as a seamless gutter.