Battenberg is a traditional English tea cake, an almond-flavored sponge cake with a marzipan frosting and apricot jam between the layers. It is highly recognizable because it is made with four (or sometimes nine) square 'layers' of sponge cake, half of which are yellow and half of which are pink; these are alternated in a checkerboard pattern, one of each color on the top layer, and then the colors reversed on the lower layer, all framed by a thick layer of white marzipan.
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Battenberg cake was invented in 1884 by the chefs of the British Royal household to celebrate the marriage of Prince Louis of Battenberg to Princess Victoria of Hesse and by Rhine. It was an attempt to invent a new, fancy cake in the general tradition of Germany (in honor of the Prince of Battenberg), and hence the jam and sponge cake, and of course, the marzipan. The result turned out to be both delicious and visually appealing, and it has become a popular cake to serve with tea (in both senses).
These days you can buy pre-packaged battenbergs (including mini-battenberg snack cakes), but many people still make their own, and recipes are readily available on the internet. Chocolate battenbergs are a common variant, but inspired bakers have made many other creative battenbergs, including Rubik's cubes, various flags (especially the Union Jack), and even a Japanese matcha/black sesame combo.
The battenberg cake is the source of the phrase Battenburg markings, referring to the bright checkered patterns found on police vehicles, ambulances, and the like in the UK.