An easy example is a set of places ordered by average daily temperature: if there are two with the same average daily temperature, the order is partial (and if not, it is total).

Another is a set of points in a circular area, ordered by distance from the centre - some will be further away than others, but there can be two different points the same distance from the centre so again the order is partial.

For a different sort of example take the set of all subsets of a given set, ordered by the "is a proper subset of" relationship. If has the original set has two or more members, this is a partial order. For example take the set {0,1}; the complete set of subsets is {},{0},{1},{0,1} and the proper subset relation has

{}<{0} and {}<{1} and {}<{0,1}

{0}<{0,1}

{1}<{0,1}

but neither {0}<{1} nor {1}<{0}