My question regards the linear model $Y = X \cdot \beta + \varepsilon$. I currently attend a lecture on linear models and I realize this issue is very basic. In our script there are plenty of equations that describe the properties of that specific linear model including it's solution $\hat{\beta} = (X^T \cdot X)^{-1} \cdot X^T \cdot Y$. My problem lies in understanding the concept.
My understanding is this: We have $n$ observations of some phenomenon, where each observation includes a part we have precise or known data for ($X$ or $\beta$) and then the observed variables we try to estimate $Y$. We try to understand the influence of $p$ factors which is why $X$ has dimension $n \times p$ and $\beta$ has dimension p.
Is my understanding so far correct? What do the indiviual variables represent? What is the meaning of $\hat{\beta}$? It is an estimator but what does it estimate and why do we care?
Edit: I guess I was naive with my question. The problem is that someone who has scarce knowledge of a topic has a hard time using proper terminology when describing their point of not understanding. Those questions were not asked to get answered one by one but more to express that I don't understand the intuition of the model. I was hoping someone who does understand it well can summarize the idea in an intuitive way.