Theta role ≠ thematic role.
The term ‘thematic role’ is used fairly often in linguistic description to label the class of things like ‘agent’, ‘patient’, ‘goal’, ‘experiencer’, ‘instrument’, and so forth. These are semantic properties that most people take to be assigned by the verb or by adpositions to the various noun and adposition phrases occurring in the clause.
In English, the subject of an ordinary transitive verb usually receives the agent role, thus in ‘I hugged her’ the subject ‘I’ is the agent that does the hugging. But if the verb is passivized then the subject gets the patient role instead, thus in ‘she was hugged by me’ the subject ‘she’ is the patient. So thematic role (agent, patient, etc.) and verb argument (subject, object, etc.) are not identical.
The concept of theta role is basically limited to theories that descend from Government and Binding (GB). Other folks here have discussed it above so I’ll skip describing it. It arose from the concept of thematic roles, hence the use of the Greek letter θ. The idea of numbering them, as done in various flavours of GB, is due to Relational Grammar (RG) I believe. RG is also the source of the terms ‘unaccusative’ and ‘unergative’, describing the alignment of subjects of intransitive verbs with either agent or patient thematic roles and hence the syntactic consequences. (I think Geoff Pullum ranted about this in his Topic...Comment column under the heading “Citation etiquette beyond Thunderdome”.)
The idea of ‘saturating’ or ‘discharging’ a theta role is that on the one hand the verb has some theta roles to hand out to noun phrases, and that the noun phrases are sitting around waiting to get a theta role. If a verb ends up with extra theta roles, or if a noun phrase doesn’t get a theta role, then the sentence is ungrammatical (or its derivation ‘crashes’ in Minimalist terms). So noun phrases have theta roles that need to be saturated, and verbs have theta roles that need to be discharged. There are a couple of other terms such as ‘fulfilled’ and the like that are also used for the same phenomenon.