Broca's aphasia occurs when a patient gets a lesion in Broca's area, located in the brain's left frontal lobe (Brodmann's areas 44, 45, and 6).
The most noticeable symptom of Broca's aphasia is a syntactic deficit characterized by labored, telegraphic speech consisting mostly of open-class content. Open-class vocabulary, or lexical vocabulary, includes words such as nouns, adjectives, and verbs, which are attached clearly to real-world meanings; it is contrasted with closed-class or functional vocabulary, which includes prepositions, conjunctions, verbal suffixes, and determiners - words that a language doesn't often gain or lose, and which are not as directly attached to real-world, concrete meanings. So a Broca's aphasic trying to say:
I went to the store with Dad.
might say:
I ... went ... store ... Dad.
The other major symptom of Broca's aphasia, which has received more attention of late from linguists and others interested in brain-language relations, is a partial comprehension deficit. Broca's aphasics can understand most of what is said to them, so for many years their condition was called simply production aphasia (as opposed to Wernicke's aphasia or receptive aphasia) and their slight comprehension deficit was ignored. But in the 1970's people started thinking about the fact that what Broca's aphasics understand and what they don't understand might shed some light on how language comprehension works in the healthy brain. After various experiments it was generally agreed that two important types of sentences English-speaking Broca's aphasics don't understand are passives and object-gap relatives.
A passive sentence is like (1):
(1) The cat was chased by the dog.
An object-gap relative is like (2):
(2) The cat that the dog chased was orange.
(It's called an object-gap relative because if you re-word it to "The dog chased the orange cat," you can see that the element ("the cat") being modified by the relative clause ("that the dog chased") in (2) is the direct object of the sentence.)
The patients' comprehension ability was judged in most experiments by a picture pointing task, wherein it was found that if a Broca's aphasic is read a sentence like those above, half the time he will point to a picture of a cat chasing a dog, and half the time he will point to a picture of a dog chasing a cat - he performs at chance when it comes to figuring out who is doing what to whom in sentences like these. (That is, he performs at chance if there are no semantic cues; if the sentence is "The cookie was eaten by the little boy," the patient can figure out that cookies do not eat little boys. Unless they are evil monster cookies. Roooooar!) On regular, active sentences ("The dog chases the cat") and even subject-gap relatives ("The dog that was chasing the cat was orange"), Broca's aphasics perform at normal or near-normal levels.
So what does this tell us about Broca's aphasics' comprehension of language, and about language in general? Well, that's a complicated question. Some think that it shows that Broca's patients lack a knowledge of argument structure; some think it shows that they don't know how to move elements in a sentence; some think they can do everything normal people can do, but with a heavier cognitive load, and so sometimes have to make tradeoffs between parsing a sentence perfectly and understanding it semantically. Understanding all of this requires more than a little bit of linguistics background; so perhaps it is a topic for another node. Until then, here is a reading list of some interesting and influential papers (most of which can be found online) on the subject, courtesy of my Neurolinguistics syllabus:
Caramazza and Zurif (1976). Dissociation of algorithmic and heuristic processes in language comprehension; evidence from aphasia. Brain and Language, volume 3, pp 572-582.
Kean, M.L. (1977). The linguistic interpretation of aphasic syndromes. Cognition, volume 5, pp 9-46.
Linebarger, Schwartz, and Saffran (1983). Sensitivity to grammatical structure in so-called agrammatic aphasics. Cognition, volume 13, pp 361-392.
Grodzinsky, Y. (1989). Agrammatic comprehension of relative clauses. Brain and Language, volume 37, pp 480-499.
Mauner, Fromkin, and Cornell (1993). Comprehension and acceptability judgments in agrammatism: Disruptions in the syntax of referential dependency. Brain and Language, volume 45, pp 340-370.
Hagiwara, H. (1993). The breakdown of Japanese passives and theta-role assignment principle by Broca's aphasics. Brain and Language, volume 45, pp 318-339. | 
