Usually found in the bathroom. Also, in certain old films, you'd see characters dip their feet in them (they would be filled with hot water) in the winter.
It's a basin.
From American Heritage (via TFD) a basin is:
An open, shallow, usually round container used especially for holding liquids.
In my experience, a basin doesn't only apply to the porcelain sink attached to the wall, but is also the (horrible!) plastic bowl that sits in a kitchen sink or under the sink in a bathroom.
It's a 'bowl'.
This link refers to a 'plastic mixing bowl' identical to the one above.
Traditionally a 'bowl' was more hemispherical than a 'basin' but that was in the days of earthenware and ceramics exclusively. Now that plastic predominates, the words are being used differently.
‘A [round] vessel to hold liquids, rather wide than deep; distinguished from a cup, which is rather deep than wide’ (Johnson). Usually hemispherical or nearly so. Historically, a bowl is distinguished from a basin by its more hemispherical shape; a ‘basin’ being proportionally shallower and wider, or with the margin curved outward, as in the ordinary wash-hand basin; but the actual use of the words is capricious, and varies from place to place; in particular, the ordinary small earthenware vessels, used for porridge, soup, milk, sugar, etc., which are historically bowls, and are so called in Scotland and in U.S., are always called in the south-east of England, and hence, usually in literary English, basins. The earlier usage remains in salad-bowl, finger-bowl (now also basin), punch-bowl, and the convivial or social bowl (see 1b).
The Ngram 'bowl/basin' indicates that 'bowl' is more common than 'basin' in AmE whereas BrE usage is equal for both.
EDIT : The Ngram devised by @Mari-LouA. gives further information.