What does it mean to have a small p value and an OR=1? As the questions says, in a logistic regression I have small p value for a variable indicating statistical significance (p = 0.001), however the odds ratio at a 95% confidence interval is 0.999 (both upper and lower bounds of the confidence interval are also 0.999). How to interpret that? I can reject my null hypothesis, but the variable has the same effect on both classes? How can I have a small p value if that is the case? 
Here is some sample output:
                      OR        2.5 %    97.5 %
(Intercept)           1.4315133 0.9037277 2.2814700
variable 1            0.9999999 0.9999998 0.9999999
variable 2            1.3925532 1.2386589 1.5640352

                   Pr(>|z|)    
(Intercept)        0.128767    
variable 1         0.00108 **
variable 2         0.00000002594048139 ***

It is unclear to me how to interpret the results for variable 1.
 A: P-values are not perfectly connected to the magnitude of an effect. For one, if you have an extremely large sample size, it is perfectly possible to have an extremely tiny effect size paired with what most would consider a small p-value.
For example, a simple correlation of 0.01 when paired with a sample size of 100K will have a one-tailed p-value of 0.00078259.
http://www.danielsoper.com/statcalc3/calc.aspx?id=44 (try it out yourself)
P-values are not an indication of effect size in and of themselves and say nothing about the magnitude of the relationship.
At some point, the binary decision process of traditional significance testing loses its utility, and you might be at that point with your data.
In terms of interpretation, the most straightforward interpretation is that you have a very, very small effect made significant on the back of an extremely large sample size. 
A: Odds ratio (OR) is equal to the estimated increase in the log-odds of the outcome per unit increase in the input variable. Hence, the value of OR is affected by the scale of the input variable. Based on your comment above, it does look like Variable_1 has a wide range (large scale).
Since the OR for Variable_1 is less than one, you can deduct the value from 1 to get 0.11, multiply that by 100, and interpret it as follows: For every 100 unit increase in Variable_1, the log odds of the outcome decreases by 11%. 
As per the apparent contradiction between p-value and OR, I agree with what @reg has mentioned in his answer above; the p-value doesn't tell you the magnitude of the effect.
