Individuals produce discourse for various purposes as part of their daily functioning. Therefore, the ability to form a discourse should be one of the main goals of functional speech-language therapy for persons with aphasia (PwA). In addition to assessing the language skills required to form a discourse, it is important to analyse how PwA form general ideas that need to be narrated. This study had two specific aims: (1) to investigate the ability of PwA, with special consideration to the stage of their recovery—the acute and the chronic phase—to form main concepts in a discourse, and (2) to examine the relationship between the number of main concepts and different types of language measures related to productivity, informativeness, and grammaticality in all tested groups. Participants included a total of 38 persons with mild and moderate aphasia (19 in the acute and 19 in the chronic phase of recovery) and 38 healthy speakers (HS) who were matched in age, gender, and level of education. In order to effectively compare the discourse produced by the groups, a single structured stimulus was used, whereby all participants were asked to orally describe a picture from the Croatian version of the Comprehensive Aphasia Test. Compared to the HS, the main concepts produced by PwA were different in number, as well as in terms of their accuracy and completeness. However, when analysing the success in the production of the main concepts with regard to the time post-stroke—acute and chronic—the difference was not confirmed, indicating great individual differences between PwA that undermine the differences on a group level. Linguistic measures of informativeness (CIUs and CIU/words) and only one measure of grammaticality (number of clauses) showed a significant correlation with the number of main concepts in PwA. When analysing correlations for PwA in the acute and chronic phases, only CIU showed a significant association with MC, confirming it as a robust measure of discourse production in PwA. PwA did not produce long and complex sentences, and they showed difficulties in the use of verbs and the marking of argument structures.