The case where two sides of a triangle are known, and an acute angle which is not the angle between the known sides. There can often be two possible triangles which satisfy all the given information, hence the name.
If you know the length of AB, and of BC, and the acute angle at A, you can construct the triangle as follows. Draw AB. From A draw a line making the required angle to AB. Place the point of a compass at B and draw an arc of a circle with radius equal to the required length of BC. Where the line from A and the arc intersect is the position of C. Unless the angle at C is exactly 90° or the given information is inconsistent, there will be two different points of intersection and hence two possible triangles.
http://yteach.co.uk/index.php/resources/sine_rule_low_sines_circle_circumscribed_triangle_area_ambiguous_page_2.html An animation showing the construction of the two possible solutions.