When you try to define something, that thing – be it concrete or abstract, chairs or thoughts – is what your definition is about.
If there is exactly one, unique such thing, your definition should, ideally, unequivocally identify it. If I were to learn that definition, I would know exactly what it is that you defined. There would be no doubt about it. I would not mistake it for something else.
In ”Naming and Necessity” , Kripke discusses the relationship between the name, a proper name of the thing or person, and that thing or person it names. Inevitably, this has a lot to do with definitions. The definition should relate the name to the thing or person. How does that happen? How could a definition do this? How should a definition be, to successfully do it?
One thing Kripke does, which is interesting here, is summarize a common view, still, on how a name refers to what it names. You can see this as an account of how a definition defines the thing being named.
”…a theory of naming which is given by a number of theses. . . [as follows:]
- To every name or designating expression X, there corresponds a cluster of properties, namely the family of those properties P such that A believes P(X) [i.e., it is true to say that X has properties P].
- One of the properties, or some conjointly, are believed by A to pick out some individual uniquely.
- If most, or a weighted most, of the properties P are satisfied by one unique object y, then y is the referent of X.
- If the vote yields no unique object, X does not refer.
- The statement, ’If X exists, then X has most of the [properties] P’ is known a priori by the speaker.
- The statement, ’If X exists, then X has most of the [properties] P’ expresses a necessary truth (in the idiolect of the speaker).
For any successful theory, the account must not be circular. The properties which are used in the vote must not themselves involve the notion of reference in such a way that it is ultimately impossible to eliminate.” 
Reference, according to the above, works by selecting properties which uniquely distinguish the named individual. To define, then, the individual, is to identify such properties at least, or all properties – those included – which describe the individual. It is to provide a description which only fits that individual.
Kripke argues this is not how reference works. The crux of his argument, to my understanding, is that people do refer to individuals, they believe and act as if they referred uniquely, yet in many cases they use names and believe reference without being able to satisfy all conditions listed above.
I agree with his conclusion, but for somewhat different reasons than those he focuses on. For one, many properties are hard to specify precisely enough, for them to be relevant in distinguishing an individual (object, person, thought) from others. Being red is a property, but there are many nuances of red, and picking the right red for the red object you are naming and defining, is a complicated matter. Not everyone sees reds in the same way, and not all red objects will consistently be observed as being red in the same way, in all conditions. So there is a practical difficulty in picking properties that distinguish. Another way to put this, is that just as we have trouble defining the thing by its properties, we have substantial difficulties defining those properties themselves. Second, suppose that there exists, as a matter of metaphysics, the possibility to know all properties of an individual (or the smallest such set which distinguishes that individual from all else). It is unlikely that I will accidentally know them all; if not, then I need to discover, or more generally do something to learn them. There is no single directory where I could learn this easily, some given universal ontology. And so, I would need to invest substantial effort to identify what picks out that individual over others. Think about all the effort that is needed to create and update biological taxonomies, medical ontologies, and really any other knowledge of classification, and it becomes clear that it is probably impractical, i.e., not feasible, for any one of us to know, for all individuals we refer to, the complete set of properties that unequivocally single out those individuals from everything else. Yet we do manage to get by in practice, as Kripke observes.
So what if we cannot successfully use properties to refer precisely?
Kripke’s proposal is, instead, that naming works through convention, and convention gets passed from person to person. Reference is part of culture, if culture is all non-genetic information passed across generations.
”Someone, let’s say, a baby is born; his parents call him by a certain name. They talk about him to their friends. Other people meet him. Through various sorts of talk the name is spread from link to link as if by a chain. A speaker who is on he far end of this chain, who has heard about, say Richard Feynman, in the market place or elsewhere, my be referring to Richard Feynman even if he can’t remember from whom he first heard of Feynman or from whom he ever heard of Feynman. He knows that Feynman is a famous physicist. A certain passage of communication reaching ultimately to the man himself does reach the speaker. He hen is referring to Feynman even though he can’t identify him uniquely. He doesn’t know what the Feynman theory of pair production and annihilation is. Not only that: he’d have trouble distinguishing between Gell-Mann and Feynman. so he doesn’t know these things, but, instead, a chain of communication going back to Feynman himself has been established, by virtue of his membership in a community which passed the name on frok link to link, not by ceremony that he makes in private in his study: By ’Feynman’ I shall mean the man sho did such and such and such and such.” 
This idea of tying reference to historical use, in which conventions pick out the individual, gets us to the topic of crowd definitions.
- Saul A Kripke. “Naming and necessity”. In: Semantics of natural language. Springer, 1972, pp. 253–355.