Do models with multiple covariates and single covariate models differ from each other? I have a binary response variable with 9 predictor variables. Lets denote the predictors $A, B, C, D, E...$
Suppose I run a model $y_i = \beta_0 + \beta_1 A + \beta_2 B + \beta_3 C + \cdots$. The results from the model tell me that $A$ and $C$ are the only significant covariates. 
Now, my question is that is there any sense (or can insight be gained) from running individual models such as $$\begin{align} y_i &= \beta_0 + \beta_1 \cdot A \\ y_i &= \beta_0 + \beta_1 \cdot B \\y_i &= \beta_0 + \beta_1 \cdot C \\ \vdots  \end{align}$$
Would I expect the individual covariate models to return the same significance as the combined covariate model?
If so, what information can running the individual models give me that I won't get from running the combined model. 
side note: I've refrained from using the word multivariate to denote the model  $y_i = \beta_0 + \beta_1 A + \beta_2 B + \beta_3 C + \cdots$. I've heard somewhere that the multivariate model actually represents multiple response variables. If I am wrong about this, please let me know in the comments.
 A: 
Would I expect the individual covariate models to return the same significance as the combined covariate model?

You mean the same p-values? No. Indeed even the sign of the coefficients could change.

If so, what information can running the individual models give me that I won't get from running the combined model.

Well, you'll have an estimate of the relationship when you don't take account of any other important predictors. In general I don't think there's much value in that, except perhaps to see how much additional variation the bigger model explains or how much the coefficients change when you do it. In some particular situations it may have some additional value.

side note: I've refrained from using the word multivariate to denote the model yi=β0+β1A+β2B+β3C+⋯. I've heard somewhere that the multivariate model actually represents multiple response variables.

Correct. With multiple covariates (/IVs) you have "multiple regression".
You didn't refrain from it in your title. Since it sounds like you meant to, I'll edit
