What am I typing on? Is it a peripheral for the ZX81 in the back, or is it a computer in its own right?
Most likely this one in a custom case. It was made/sold by now defunct Maplin as one of their early own computer products in 1981. It was available as PCB only, Kit or Assembled product. Its larger cousin, the 'Full Size Keyboard' was described in detail in their Maplin Projects Book Three of 1983 on p.28.
Excerpt from the Maplin price list of Nov/1982:
XH58N Keytop Print ZX81 25 XG17T ZX81 Keyboard Case £4.95 GA83E ZX81 Ext Kyboard PCB £2.95 LW72P ZX8I Keyboard Kit £19.95 XG22Y ZX81 Keyboard £29.95
According to the price list it was listed on page 243 of the catalogue (p.303 for the 1984 catalogue).
The Computer is most likely a genuine ZX81 with Sinclair's ZX 16K RAM 16 KiB RAM expansion and the ZX PRINTER. A configuration much like on this Wiki-picture - sans the keyboard of course.
It looks like a Timex Sinclair 1000 with a modded keyboard. Possibly by Memotech.