Use with or without an object refers to a verb's transitivity.
A verb with an object -- specifically, a direct object -- is used transitively and is in this circumstance called a transitive verb. The direct object is the noun or noun phrase which is affected by the action:
Mary walked the cat. ... cat is the direct object.
John is studying English. ... English is the direct object.
Don't hassle me. ... me is the direct object. The sense is "harass".
A verb without an object is used intransitively, and is called intransitive:
Mary walked to town ... Mary doesn't affect another person or thing.
John is studying. ... John doesn't affect another person or thing.
Boys, don't hassle. ... Although the boys are presumably affecting each other, they don't affect anyone else. The sense is "quarrel".
Some verbs take two objects, a direct object and an indirect object. The indirect object is the noun or noun phrase which receives the action or for whose benefit is done. These verbs are used bitransitively.
John gave Mary a book. ... book is the direct object, Mary is the indirect object.
Bob told me his story. ... story is the direct object, me is the indirect object.
Note that object in all of these is syntactic category, not a semantic category:
The story was told by Bob. ... Although story is affected by the action, it is not the object of the verb but its subject, because this sentence is in passive voice.