This portrait is by Hans Holbein the younger, circa 1527, and it is generally referred to as "Lady with squirrel and starling" since it is a portrait of a lady with a squirrel and starling. The starling is in the background, just like the leaves there and not that much of an eye-catcher.
The squirrel is another matter. It sits in the lady's arms, obviously chained in some manner, and the lady is holding the chain. This was a time when people kept pets that we would consider right out strange - but then again, the difference between having a squirrel and a bunny might not be that great. It is after all a matter of habit.
The lady herself is somewhat of a mystery. As late as 2004 it was suggested that she would be Mrs. Anne Lovell - which would fit with the date of the painting and the squirrel was a common symbol for the Lovell family and was used in other depictions. Holbein often had animals depicted in his portraits.
The lady, Anne Lovell or someone else, is dressed in a rather sombre dress. It is black, or possible dark blue - it is a bit hard to tell just from pictures of the painting, but in any case it is not mourning clothes. Over her shoulders she wears a white shawl, probably of linen, and over the hair she is wearing a white cap. The form is one that reminds you very much of the typical Tudor-cap, with the angles at the back of the head - but the thick structure is not quite so common. Is it perhaps something for winter-wear?
In 1992 the painting was bought by the National Gallery in London.