In penalized regression, 'L1 penalty' and 'L2 penalty' refer to penalizing either the L1 norm of a solution's vector of parameter values (i.e. the sum of its absolute values), or its L2 norm (its Euclidean length). Techniques which use an L1 penalty, like LASSO, encourage solutions where many parameters are zero. Techniques which use an L2 penalty, like ridge regression, encourage solutions where most parameter values are small. Elastic net regularization uses a penalty term that is a combination of the L1 norm and the L2 norm of the parameter vector.
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