the unknown function
In ordinary algebra, when we 'solve for x' we are hunting for a number. In differential equations the thing we are hunting for is not a number at all — it is a whole function, an entire curve. This is the single biggest mental shift the subject asks of you. The 'unknown' is the rule that gives the output value at every possible input, all at once.
Precisely, the unknown function is the dependent variable regarded as a yet-undetermined function of the independent variable — usually written y(x) or y(t), often shortened to just y. The differential equation is a set of conditions this function must satisfy; solving means finding the actual formula or curve y = f(x) that fits. So in y' = k y, the unknown is the function y(t), and the answer turns out to be y(t) = C e^(kt) — a function, with C still free. Until we solve, y stands for a function we do not yet know, the way x once stood for a number we did not yet know.
Keeping this in mind keeps you honest about what an 'answer' looks like. A solution is never a single point; it is a function defined over an interval, and usually there is a whole family of them until extra information narrows things down. This is also why we can verify a candidate by substitution: plug the proposed function and its derivatives back into the equation and check that it holds — because the unknown was a function, the test is whether that function works everywhere.
For y' = k y the unknown is the function y(t). A correct answer is the function y(t) = C e^(kt), not a number — for each choice of the constant C it gives a complete curve of values for all t.
The thing you solve for is a function, supplied as a formula in the independent variable.
Do not confuse the unknown function y(t) with its value at one instant, y(3). The equation determines the whole function; an initial condition then fixes its value at one point.