In syntax, an argument is a word or phrase that has a syntactic relationship with the verb of a sentence. In layman's terms, the arguments are the things (often people) which are performing the verb, or being acted upon. Subjects and objects are the two most common arguments in sentences.
English divides its arguments into Subjects, Objects, and Indirect objects at the syntactic level. This is reflected in the word order; the subject comes first (I hit her), the object second (I hit her), and indirect objects can appear either after the object with a preposition (I gave the book to her) or without a preposition phrase before it (I gave her the book). The subject is generally though of as the active argument, "the one performing the verb," while the object is the one being affected by the verb - but this is inaccurate in some cases; in the sentence "I was hit by a bus," I is still the subject, even though it is clearly the thing being hit. (This is called passive structure in English.)
Related to arguments and cases are theta roles, which describe what an argument is actually doing or having done to it, rather than its syntactic position in the sentence. For instance, in the passive construction in English seen above, "I" is in the subject slot, but it is not taking an active role - it is in fact the patient (thing which is acted upon) of the sentence, whereas the agent (thing which acts upon something else, generally with volition) is the bus.