My understanding of the FDR is that it is synonymous with the Type 1 error or rejecting the null when the null is true. However, I am not certain how this relates to linear models. I read that false discoveries occur in high dimensional logistic regression models. What exactly is the false discovery? Is it a positive estimated $\beta$ coefficient when it should be zero for the MLE of the coefficient's?
A false discovery is a type 1 error - Basically when your coefficient is statistically significant, even though there is no relationship between the two things in question. False discovery rate is a way of adjusting your p-values in a multiple-testing scenario. The FDR described by Benjamini, Hochberg and Yekutieli (2001) basically controls the proportion of your discoveries (rejected null-hypothesis) that are false, that is, (type 1 errors) / (type 1 errors + correctly rejected hypothesis)
To understand this better, you have to first understand the problem with multiple testing: When you try to (statistically) find something for long enough, you will find it with certainty. That's just the nature of randomness, eventually you will find a null-hypothesis that can be rejected at any chosen confidence level.
The same applies to linear regression, when you have 1'000'000 variables in your model, on a 5% confidence level, you would expect 50'000 variables to have a statistically significant coefficient, even if all are false. Multiple testing adjustments, such as the (conservative) Bonferroni correction, or the FDR, can be used to deal with this problem (in various ways).
So basically, what false discovery rate does, is it allows you to control the fraction of "discoveries" (non-zero coefficients) that are wrong. When you make many discoveries, you will make more errors, using FDR, when you make less discoveries, you will make less errors.