I'm taking Regression Analysis right now, so I'll try my best to explain it.

In regression analysis, as you said, we try to create an equation to relate the dependent variable (y) to the dependent variables(x1, x2,...,xn).

It's called the dependent variable because it's value depends on the value of the independent variable (whose value doesn't depend on anything).

In a simple model, there is only 1 independent variable, while in multiple model, there are many.

Simple model -

multiple model -

where is a parameter.

Example

We want to create an equation to predict the crime rate to different facters. In other words, we want to relate the dependent variable ( y - crime rate) to the independent variables (x - crime factors)

X1 - Location

X2 - Status of the economy

X3 - Unemployment rate

So the multiple regression model in this example would be

We just started non-linear, so I don't know much about it. The previous examples are linear, while the following isnt:

is also linear, which is what you quoted from Wikipedia "but need not be linear in the independent variables". It has to be linear in terms of the parameters.