Use of determinants

Determinant is a value associated with a square matrix.

Gee, that tells you a lot, does it not? Let's try to make it more intuitive "why it exists" and "how it works".

First thing you need to know is that it doesn't need to exist. Nor do matrices. Nor does anything. But it's a helpful concept which helps us solve problems which reoccured a lot of times and it was deemed prudent to give it a name. Like it was prudent to construct the concept of matrices to assist with systems of linear equations. See, everything mathematicians do is to resolve problems efficiently and with style. Yes, we do love style. And through our investigation, some values and calculations will prove to be consistent and very useful. And very meaningful context-wise. And we'll give them names and make them our friends. Or tools.

In the early days, before mathematicians discovered the elegance of matrices for resolving systems of linear equations, determinants were linked to systems of linear equations themselves. Their name is actually a give-away - it literally determines whether the system has a unique solution (nor over-constrained nor under-constrained). And the math showed that this value has to be everything but $0$ to hold true. To have a unique solution. After these first baby steps, many mathematicians added to the foundation of the original theory and the concept of the determinant flourished as people solved more and more problems. I'll try not to scare you with details, you can investigate on your own.

A determinant is used in many context-specific ways. As one of the comments said, by Joe Johnson, they show up in multivariable calculus. In the context of its earliest usage, they were an indicator whether a system of linear equations has a unique solution. The condition? The value given the name of "determinant" has to be nonzero. As matrices became more useful over time and got many new upgrades, speaking figuratively - for example - we can represent linear transformations ( a set of linear equations which respond to a few rules and transforms vectors from one coordinate system to another, for example). Now, the pretty part of matrices in the geometric context is that if you multiply a point which you've transformed with a matrix $A$, you can undo that with an operation with its inverse $A^{-1}$. But there are cases when a matrix simply doesn't have a friend matrix which when multiplied together yields the n x n identity matrix - the equivalent of of number $1$.

Mathematicians found out that the good old concept of a determinant can be used as an indicator whether an matrix has an inverse. If it's 0, no inverse - sorry.

This is a simplification to get your feet wet. Just understand what the primary idea behind every concept in mathematics is. We don't pull them out of our behind, they knock on our doors when we work on problems. A fool shows them the way out by ignoring them, a wise man accepts their offerings. To be a bit poetic.

Hope this answers your question which was a bit... Well... Ambiguous. If you want more, clarify and the community will jump in to assist. When you leave it very coarse, a book can be written. And math.SE is about specific problems that can be addressed. Whether a proof, intuitive understanding or simply a quick answer.

I commend your chase of a firmer intuitive understanding. Sometimes, it is difficult, but don't give up.

Determinants can be used to see if a system of $n$ linear equations in $n$ variables has a unique solution. This is useful for homework problems and the like, when the relevant computations can be performed exactly.

However, when solving real numerical problems, the determinant is rarely used, as it is a very poor indicator of how well you can solve a system of equations, and furthermore, it is typically very expensive to compute directly. Other quantities (such as singular values) provide better indications of 'solvability', and other techniques (Gaussian elimination, QR decompositions, etc.) are better for solving systems of equations.

The determinant also gives the (signed) volume of the parallelepiped whose edges are the rows (or columns) of a matrix. I find this interpretation to be the most intuitive, and many standard results for determinants can be understood using this viewpoint. (However, I have rarely had a practical need to compute volumes using determinants.) The volume interpretation is often useful when computing multidimensional integrals ('change of variables'). It is also useful for understanding (or defining) the 'cross product' in physics or mechanics.

The determinant is a very useful theoretical tool, whose applications extend well beyond matrices of real or complex numbers. However this may not be apparent at the calculus level.