The Greek word for "heart" is kardia (καρδία) (noun sg fem). It means the physical organ, the seat of emotions (especially passion, rage, and anger), and "inclination", "desire," "purpose", "mind", "the pith (in wood), and "the deep (of the sea)." Jesus uses it to describe the feelings that motivate a person's actions (see Matthew 15:19). It is best understood in the context that Jesus describes the various components of humanity described in this article here.
The word for "heart," kardia, is used thirty-seven times by Jesus. Making it one of the most common components of our human nature that he mentions. It is closely connected to the word meaning "spirit." Sometimes this word is used like we use the word "heart" to mean the "insides" of something ("heart" of the ocean or earth), but when it is used to refer to people, it is the seat of emotions. These emotions do clearly generate thoughts (Matthew 15:19) so feeling comes before thought as a cause. Note that in this verse, Jesus views the human heart as inherently flawed. It gives rise to "worthless thoughts." The heart is the motivation for thought, the cause of thought. It is not the thoughts themselves. The spirit experiences thoughts in the mind through the urgings of the heart.
The Social Heart
One interesting aspect of this word is that Jesus often uses the singular, "the heart," to describe the thoughts or feelings of a group. For example, he uses the plural possessive "your" with the singular "heart" in Luke 24:38. Though the phrase is translated as "in your hearts" in the KJV, the Greek ἐν τῇκαρδίᾳ ὑμῶν; literally means "in that heart (singular) of yours (plural). In Matthew 15:8, we can see that he gets this usage from the Septuagint, so this may have been a general usage of the concept of "heart" in his culture. So its sense would have been more of a shared "emotion" or "source of emotion" than the physical heart. So "heart" describes a shared mindset.
This use of the singular for the group's "heart" indicates that Jesus recognized our feelings as arising from something greater than ourselves. "Heart" is connected both to mind and spirit. "Mind" is formed by our living within a social order, which is shared. Our feelings are, at least in part, learned from our environment. Similarly, our feelings are at least in part, generated from, "Spirit," divine influence, which is also shared. We are not responsible for them or the thoughts they generate. We are only responsible for our actions. The choices made by our individual spirits despite the urging to our social hearts. I am reminded of Huck Finn's battle conscience between what society taught him about slavery being good and right versus the urgings of his own spirit.
Relative Greek Concepts
Kardia is also related to an important Greek concept Jesus only uses once, stethos ( στῆθος), which means "breast". The same concept is more frequently called thumos, (θυμός), which literally means "chest". This is another Greek concept of "soul," "spirit," "feeling and thought," and especially "of strong feeling and passion." Logically, the breath of pneuma goes into the breast or chest and from there to the heart. So, the heart is where spirit and mind are united. Lower animals are driven more by the belly. This Greek concept of thumos or "feelings of the chest" or, as we might say, feelings of the heart. These included the love of family and country, hatred of enemies, anger, passion for learning, etc. Jesus seems to use kardia to to describe all of this. The kardia/heart might be described as the desires of the mind in this context.