I received these output from R:Estimate, Std. Error, z value, and Pr(>|Z|). I know z value = Estimate / Std E, but what exactly is "std. error" and Pr(>|z|) mean?
The "std.error" is the standard deviation of the coefficient point estimate in the GLM. It is a measure of uncertainty about this estimate - if it is too large, then you have a coefficient point estimate calculated with a lot of imprecision.
The "Pr(>|z|)" is the so called "p-value" of the test for whether the coefficient point estimate is significantly different from 0. Intuitively, it tells us if our point estimate has been calculated precisely enough to distinguish it from zero. We typically define "precisely enough" using the $p < .05$ maxim.
All other things held constant, the larger the "std. error", the larger the p-value.