What's inside a 'cell' can vary per manufacturer, but the Xilinx definition (using this manufacturer as an example, as these are the devices that I'm familiar with) is one four-input look-up table, and one register. Xilinx devices are made up of a number of 'slices', and these contain a number of functional elements. These might include:
- Look-up tables
- Registers
- Multiplexers
- Logic for use in carry chains
- etc
As an example, a Spartan6 LX4 has 600 slices, and the marketing material claims that this is equivalent to 3840 'logic cells'. You can look in the user guide for a device to determine exactly what is contained inside a slice.
In addition to this, there are other resources such as multipliers, memories, PLLs, etc.
I suppose you could say that one logic cell can perform one operation, but a single cell is only capable of very simple operations, for example an AND gate, 2:1 multiplexer, etc.