It's quite simple. First, we note that the function

f(x) = exp (-1/x) |x| < 1
0 x >= 1

is infinitely differentiable and never negative. Likewise, in n dimensions, the function

g (x_1, ... x_n) = f ( sqrt{x_1^2 + ... sqrt x_n^2)

is infinitely differentiable and differs from zero only when the point lies in the unit ball. Now, if M is a manifold then, by the paracompactness property, we can cover M with a (possibly infinite) number of balls such that any point is contained in a finite number of these balls. To each ball B, we associate a function like g_B as above (which is stricltly positive inside the ball and zero outside). Sum these functions. Because of our paracompactness property, this sum converges trivially for any point p of M, and define h(p) to be the sum of g_B (p) ofer all balls B in the cover. Note that h(p) is strictly positive for all p. Thus g_B / h exists. The collection of all g_B / h's for all B is your partition of unity.

Surely we can do better than a circular, one-word semantic definition here....try Kelley, (even Wikepedia!)....
To quote someone: 'incomplete truth is better than no information'.

Best wishes.