Is linear regression the same thing as ordinary least squares in SPSS?

Yes, although 'linear regression' refers to any approach to model the relationship between one or more variables, OLS is the method used to find the simple linear regression of a set of data.

Linear regression is a vast term that just says we are finding a relationship between the dependent and independent variable(s), no matter what technique we are using.

OLS is just one of the technique to do linear reg.

Lets say,

error(e) = (observed value - predicted value)

Observed values - blue dots in picture

predicted values - points on the line(vertically below to the observed values)

The vertical lines below represent 'e'. We square them -> add them and get total err. And we try to reduce this total error.

For OLS, as the name says (ordinary least squared method), here we reduce the sum of all e^2 i.e. we try to make the error least.