There are different brands of grocery stores in Slovakia, you can find little ones but be ready that they are more expensive, but convenient.

Biggest are LIDL, Tesco and Kaufland (Kaufland has only few stores in Bratislava).
Tesco is the only one which operates online as well and has British language mutation on the online shop :). Lidl has online shop only for nongrocery products.

It is really worthy to mention also Delia - smaller cozy store concept, many times with coffee corner as well. They do really fast delivery.