What is the area of this polygon?

Mathematica, 13 bytes

Area@*Polygon

Octave, 9 bytes

@polyarea

Inputs are a vector with the x values and a vector with the y values. This works in MATLAB too.

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JavaScript (ES6),  69 67 47  44 bytes

^{Thanks to @Rick for noticing that we don't need the absolute value if the vertices are guaranteed to be sorted in counterclockwise order and for suggesting to take a flat list as input, saving 20 bytes!
Saved 3 more bytes thanks to @Shaggy}

Takes input as a flat list of vertices, including the last vertex.

f=([x,y,...a])=>a+a?x*a[1]/2-y*a[0]/2+f(a):0

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How?

This is based on the formula used in many other answers. For \$n\$ 0-indexed vertices:

$$area=\left|\frac{(x_0y_1-y_0x_1)+(x_1y_2-y_1x_2)+\ldots+(x_{n-1}y_0-y_{n-1}x_0)}{2}\right|$$