I'm dealing with oncology patients so it would be nice to know whether to use univariate or multivariate cox regression. I have some books on survival analysis but they don't elaborate the academic difference and interpretation of results regarding both methods.
I think that many people who use the words "multivariate regression" with Cox models really mean to say "multiple regression." (I will confess to having done that myself; it's common in the literature.) "Multiple regression" means having more than one predictor in a regression model, while "multivariate regression" is a term perhaps better reserved for situations where there is more than one outcome variable being considered together. In a Cox regression you are typically modeling just a single outcome variable, survival of some sort.
If you are preparing results for publication in a medical journal, the editors and reviewers will typically expect to see a table of single-variable relations of predictor variables to outcome (your "univariate" regressions). These single-variable relations, however, are seldom very informative due to relations among the values of the predictors and potential interactions among the predictors with respect to outcome.
These issues can be handled by Cox multiple regression, which gives you the best chance of evaluating each of the predictors with all the others taken into account, and which allows directly for testing of interactions. You have to be careful not to evaluate too many predictors together in a model, however. A useful rule of thumb is that you should limit your analysis to no more than 1 predictor per 10-20 events (recurrences or deaths in oncology) in a standard Cox multiple-regression model.
Note that there can be a true multivariate Cox regression that evaluates multiple types of outcome together (e.g., both recurrence and death times in cancer studies), or that treats multiple events on the same individual with multivariate techniques, as in standard multivariate linear regression. This paper is one often-cited reference, in case that is what you actually mean. But in my experience, I think most people in the clinical literature say "multivariate Cox regression" when they really mean "Cox multiple regression."
It would be wise to get some more direct advice from a local statistician, as there are many issues that need to be considered in building a reliable survival model. Working with an experienced practitioner can also be an efficient way to learn for yourself.
You should opt to do multivariable cox regression analysis (Not multivariate). As rightly point out by @EdM multivaraite means having more than one outcome variable, whereas, in survival analysis you have only one outcome variable, i.e. time-to-event of interest. Since, in oncology the group of patients under study, in most cases, is heterogenous, my advice would be to conduct multivariable analysis.