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From the abstract to the study titled "Association between the antioxidant uric acid and depression and antidepressant medication use in 96 989 individuals":

We examined plasma levels of the antioxidant uric acid in 96 989 individuals from two independent cohort studies. Logistic regression and Cox proportional hazards regression models were multivariable adjusted for age, gender, alcohol, smoking, income, body mass index, C-reactive protein, hemoglobin, triglycerides, cardiovascular disease, diabetes, and intake of meat and vegetables. Results were performed separately in each study and combined in a meta-analysis.

What is the meaning of this in simple terms?

As I understand, the authors used "logistic regression models" and "Cox proportional hazards regression models", and these models were "multivariable adjusted".

But what does it mean - "multivariable adjusted"? I would like to have a simplified explanation as an aid to translation.

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On the one hand, we don't know what the authors did. So if you want a definitive answer, I'd recommend you ask the authors themselves.

On the other hand, this language is consistent with straight-up logistic or Cox regressions that included all the covariates that were "multivariable adjusted for", plus the ones that were of primary interest. I often see German psychologists use similar language in such a case. Note that the authors are almost certainly not English native speakers, to judge from their names and affiliations.

Adjusting and controlling for variables is discussed at length here.

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