Your calculator (and Alpha) are correct: $-1^2 = -(1^2) = -1$ whereas $(-1)^2=+1$.
When evaluating expressions with multiple operations you have to follow the proper order of operations (order of precedence).
Quickly: Exponentiation then Multiplication/Division then Addition/Subtraction.
The reasoning behind this ordering has to do with the complexity of each operation (which ones are built from which other ones).
The most basic operation is succession (let's use "S" to denote successor). Then $S(n)=n+1$. Concretely $S(0)=1$, $S(1)=2$, etc. The inverse operation is declination (go down one): $D(n)=n-1$ so that $D(5)=4$ etc.
Addition is built from repeated succession: $3+2 = S(S(3))=5$ (start at $3$ and then go up twice). Subtraction is undoes addition. It can be realized as repeated declination: $3-2 = D(D(3))=1$ (start at 3 and go down twice).
Next, multiplication is built from repeated addition: $3 \cdot 2 = 3+3=6$ and division is built from repeated subtraction: $12/4 = 3$ since $12-4-4-4=0$ (three 4's to eliminate 12).
Finally, exponentiation is built from repeated multiplication: $2^3 = 2 \cdot 2 \cdot 2 = 8$.
Thus increment and decrement repeated give addition and subtraction. Addition and subtraction repeated yield multiplication and division. Multiplication repeated yields exponentiation.
This leaves us with our standard order of precedence.
So finally, $-1^2$ involves an exponentiation and a subtraction (negation is just a special case of subtraction -- subtract from zero). So exponentiation is "more important" (higher up on our hierarchy of operations) so it's done first: $1^2=1$. Then comes negation: $-(1^2)=-1$.
Of course, parenthesis override precedence (we must compute expressions within parentheses first), so $(-1)^2$ says to negate 1 first: $-1$. Then square: $(-1)^2 = (-1) \cdot (-1) = 1$.