The semantic role, or thematic role, or theta role (I think there might be a few other names I'm forgetting right now, for that matter) is an important tool in linguistics. See, languages mark the role that a noun phrase plays in a sentence in all sorts of different, wacky ways. A noun phrase in a language like English could be the subject of a sentence, or an object, or it could occupy some other role - for instance, the sentence I put the book on the table has three arguments: a subject (I), an object (the book) and a destination for whatever is being put (on the table). The trouble is that notions like "subject" and "object" don't translate real well across languages.

Why don't they? Well, often the most natural translation of a sentence will assign the same actors to different grammatical functions in different languages. For instance, the sentence He likes potatoes translates into Spanish as Le gustan las patatas. The meanings of the sentences are the same, but the Spanish sentence uses potatoes as the subject of the verb and le ('to him') as the indirect object, while in English he is the subject of the sentence, and potatoes is the direct object.

Semantic roles provide a set of vocabulary that can be used to describe the role of each participant independent of the grammatical role they play - indepedent, that is, of how they're marked by each language - strictly in terms of what they mean. So in both languages, potatoes play the semantic role of stimulus, while he is the experiencer.

Semantic roles are useful in examining a single language as well. Most languages have different ways to cast the same sentence. For instance, virtually all languages can form passives, in which the direct object of the more basic sentence acts as the subject (in English, as with many but not all languages, the subject of the original language can be optionally indicated in the passive, as the object of a preposition.) So a sentence like Tasha bitch-slapped Arturo has Tasha as the subject and Arturo as the direct object. But it can be recast as Arturo was bitch-slapped by Tasha, and the meaning of the sentence doesn't change much. However, in the new sentence, Arturo is suddenly the subject, and Tasha is the object of a preposition.

Since both sentences have the same meaning, the semantic role of each participant stays the same - but the grammatical roles have changed. In both sentences, Tasha occupies the semantic role of agent - meaning the participant that chooses to initiate an action, and Arturo is the patient - that is, the person who is acted upon. So using semantic roles gives us a way to describe the underlying role played by a noun, independent of the ways the sentence might be recast or how it might be rendered when translated to a language that assigns grammatical relations differently.

So it's important to recognize that semantic roles are semantic (of course) and not syntactical or morphological - that is, they depend on the meaning of the sentence and not necessarily on the grammar, so nouns in a paraphrase or a translation of a sentence will possess the same semantic role even if the grammatical relations don't match up. Of course, semantic role is not independent of grammatical relation. For instance, in English, in transitive sentences (those with a subject and a direct object), the subject of the sentence is usually an agent, and the direct object is usually a patient. But such correspondences are only tendencies; semantic roles are useful precisely because they allow us to step away from the sometimes puzzling ways that languages assign subjects, objects, and other grammatical roles.

One bit of theory surrounding semantic roles is that in any sentence, each noun phrase will occupy exactly one semantic role. Further, one semantic role will not be shared by two different noun phrases (notwithstanding coordinations, when two nouns are joined into a single noun phrase with a conjunction, e.g. Chloë and Jacqueline ate at the new Ethiopian restaurant downtown, in which Chloë and Jacqueline form a single noun phrase playing the role of agent.) In fact, this correspondence has been identified by some as a language universal, although it's possible that this is just a result of how linguists have analyzed sentences.

Since semantic roles are identified by judgments about what role noun phrases occupy in a sentence, and are not clearly marked by languages the way grammatical relations like subject and object are, the number of different semantic roles varies according to different linguists. The list below (taken, with some modifications, from Whaley, though the examples are mine) captures some of the most commonly recognized semantic roles.

Agent: an entity causing an action.

Paula baked special brownies to bring to the party.

Patient: an entity affected by an action.

Max broke all of his ex-boyfriend's CDs.

Stimulus: something provoking a cognitive state.

James enjoys masturbation.

Experiencer: an entity receiving a cognitive experience.

Darla was strangely fascinated by the carnage resulting from the accident.

Comitative: something that accompanies another entity.

Marie wandered through the park with her pet wallaby.

Instrumental: a tool used to perform a task.

Carly conquered Madagascar with her army of giant robots.

Recipient: the entity receiving an object.

Janine sent a box of laxative-laced chocolates to Charlie.

Source: origin of motion.

Angelo hurried out of the hotel room after discovering that the hooker had a penis.

Goal: end point of motion.

Vanessa got out of the taxi and staggered drunkenly up to her apartment.

Locative: point in space where an action occurs.

Jared got an unpleasant surprise in the doctor's office when he received his herpes diagnosis.

Temporal: point in time when an action occurs.

Last Thursday, Sonja bought her first vibrator.

Purposive: the reason for an action.

Charlotte bought capers and fresh basil for dinner with her new lover.

Benefactive: the entity benefitting from an action.

Antonio purchased flavored lube to surprise his boyfriend.

This list is pretty expansive compared to those postulated by some authors. Because what constitutes an independent type of relation between two noun phrases depends on how categories are defined, many of these aren't recognized by some linguists as separate roles. The total number of semantic roles in different analyses varies between around eight and twenty depending who's defining them.

References

Whaley, Lindsay J. 1997. Introduction to Typology: The Unity and Diversity of Language. Thousand Oaks, CA: SAGE Publications, Inc.
Comrie, Bernard. 1989. Language Universals and Linguistic Typology. Chicago: University of Chicago Press.