What is a multivariate logistic regression There are a lot paper in medical researches using multivariate logistic regression. I am wondering if the multivariate logistic regression is just the mixed effects logistic regression or something else.
It just a screen shot from the Google scholar since 2014 below.

 A: In statistics, multivariate and multiple mean two different things all together. In a regression model, "multiple" denotes several predictors/independent variables. On the other hand,  "multivariate" is used to mean several (2 or more) responses/ dependent variables. To this end, multivariate logistic regression is a logistic regression with more than one binary outcome. For example including both HIV status (positive or negative) and  Condom use(Yes or No) as response/outcome in the same logistic regression model. Both responses are binary (hence logistic regression, probit regression can also be used), and more than one response/ dependent variable is involved (hence multivariate). NOTE In multivariate analysis, there should be some correlation between the responses used in the model. Otherwise, separate logistic regression models should be fitted for each response. In the above example with HIV status and Condom use as dependent variables, there should be some within subject correlations between HIV status and Condom use.
As opposed to multivariate logistic regression, a multiple logistic regression is a logistic regression with only one response but several predictors. 
For example predicting HIV status (Positive or negative) using the number of sexual partners, and the practice of safe sex as possible independent variables. Here, only one response is involved (HIV status). However, there are two or more (in this case only two) predictors / independent variable namely, 1.number of sexual partners, and 2.practice of safe sex
A: There is no reliable answer to your question in terms of standard usage in the medical literature. The paper cited by @DirkHorsten, in a comment on another answer here, examined articles published over the course of a year in a single high-quality public health journal that presumably has statistical expertise in its reviewing practices. Sometimes the word "multivariate" represented longitudinal data, sometimes it simply represented multiple predictors in a model with only one outcome variable and no mixed effects.
I'm afraid that you will, at least for now, have to read each paper to know what the authors meant by "multivariate."
A: 'Standard logistic regression' (part of generalised linear models)
The logistic regression can be the 'standard' logistic regression with fixed coefficents, so in the univariate case (for simplicity I take one explanatory variable, but the reasoning holds also in the multivariate case), the logistic regression tries to predict the probability of 'success' conditional on a given value of the explanatory variable: $P(y=1|_{X=x})=\frac{1}{1+e^{-(\beta_1 x + \beta_2)}}$. 
If you assume that your coeffcients are fixed then you have the logisitic regression as explained by @Enrique (+1). 
Mixed effects logistic regression (part of generalised mixed effect models)
But in some cases it may be interesting not to assume fixed $\beta_i$ e.g. because you think that for each participant in a survey the intercept is different (see an example in the case of a linear model in my answer on How to account for participants in a study design?). If you assume thet e.g. $\beta_2$ is random , then you can estimate these coefficents for this 'mixed effects logistic regression model' . 
The estimation of the parameters $\beta_i$ is a bit more complicated but implemented in most statistical software. The interpretation of the estimated parameters is harder however. 
A: You may want to take a look at "Bahadur Model".
In Bahadur model, "multivariate" in "multivariate logistic regression" means multiple binary dependent variables, which may be correlated at second or higher order. See references from Verbeke and Molenberghs for a description of this model.
Best,
Rolando
A: Multivariate logistic regression is like simple logistic regression but with multiple predictors. Logistic regression is similar to linear regression but you can use it when your response variable is binary. This is common in medical research because with multiple logistic regression you can adjust for confounders. For example you may be interested in predicting whether or not someone may develop a disease based on the exposure to some substance. In that case you may use a simple logistic regression model but it may be the case that the disease has no relation with the substance but with age. In that case you can include in the model both: substance exposure and age so you can analyze if age is a confunder, i.e., the disease may be correlated with age but not with the substance. You can also use multiple logistic regression to increase your prediction power by adding more predictors instead of just using one.
