When people refer to logistic regression it's typically binary logistic regression so that the response variable only has two possible outcomes. With ordinal logistic regression the response can be a bit more generalized and as long as there is some sort of way to "order" the possible outcomes you can use this method. So you could have 3 or more possible outcomes with ordinal logistic regression (think something like "do not agree", "somewhat agree", "completely agree") whereas with what people typically refer to as "logistic regression" you can only have two outcomes.
There are actually two types of logistic regression with more than two levels on the dependent variable. Ordinal logistic regression assumes the levels can be ordered, as with the likert example Dason gave. Multinomial logistic regression does not make the assumption that the levels can be ordered. An example would be a dependent variable where you could be Christian, Jewish, or Muslim. There is no inherent ordering in the variable.