This is called controlling. Simple and multiple regression results may differ. What you report depends on the research question you have. Generally speaking, multiple regression results are important.

In your multiple regression you have at least three variables: two predictors (X1 and X2) and an outcome (Y).

In a simple regression X1 predicts Y so X1 and Y are correlated. What happens if you add X2 to the model? It depends on the correlation between X1 and X2 and the correlation between X2 and Y and between X2 and the residuals of the model predicting Y from X1.

If X2 only improves the overall model fit if it explains new variation in Y (i.e., it correlates with the residuals of the simple model). If it doesn't improve overall prediction but is correlated with X1 and Y then the estimated effect of X1 will decrease and may become non-significant. This is because X1 doesn't uniquely explain Y (it overlaps in variance explained with X2). In these situations you'd expect the overall model to still be significant (though adding predictors can decrease power in small samples by eating up error df) - indicating that X1 and X2 jointly explain Y, but you don't have the statistical power to adequately disentangle their effects.

This is an example of collinearity of the predictors. With more predictors you get even more complex patterns owing to multicollinearity unless you are lucky and your predictors are independent of eachother.

My first impression is that your X's are correlated, the outcomes you are describing indicates are frequently a symptom of multicollinearity

My Dear

I agree with  all  above answers , necessary test the correlation between X's varaibles  if yes in this case use some satatitical methods to solve this problem(Hitero.)

with my all thanks

Many thanks to all of your answers. Very helpful!

Yes, the above responses are in line. Notice that if you use reference cell  coding scheme, the effect of the subsequent covariates might be masked by other covariates if there is evidence of correlation among the covariates. Thanks

All of the above answers provide useful information for your first question, which is to address why it happened.  For your second, yes, you should report both the results of the simple and the multiple regression, as well as the full correlation matrix between your predictors and outcome. 

   Note - when you stated that "While when I performed a single multiple regression, the significance was gone."  The above answers presume that the significance you are describing is for your primary predictor (let's call it X1).  But you could mean that none of the predictors were significant (at minimum both X1 and X2 were not uniquely related to your outcome), and/or that your overall R-square was not significant (that would be very very odd and worth considering what might have gone wrong analytically).    It would be helpful if you might clarify what, exactly, was not significant in your multiple regression.

Thanks for the reply Julia B.That's what I'll do, I'm going to report both of them and explain the correlation.

Yes the overall R-square was still significant but the primary predictor wasn't. After checking correlation I found out that it is due to multicollinearity as suggested by the above answers.

The problem you are asking occurs due to multi-collinearity problem in your data set. You should verify the high correlation between independent variables in the model. In other words, your independent variables should not be highly correlated.

If single variable gives low R square with significance and multiple regression gives high R square but insignificance, which model is preferred?

I´ll go with theoen variable & significant, since it states that there at least that one variable in your Xs explain part of the variability of Y. High R-square can be deceiving

Model 1-Regression run with X1 gives low R square but significance

Model 2-Regression run with X1 and X2 gives high R square but insignifance for X1and significance for X2

which model should we prefer. Should we go for high R square although the X1 is insignificant or Model 1 with Low R square but X1 is significant

Anant Pandey it depends on your purpose from fitting a regression model. If your goal is to understand how the variables work together to explain Y, then statistical significance is recommended. However if your goal is to predict Y from X1 and X2, then high adjusted R square is recommended regardless of statistical significance and collinearity (this, however, increases possibility of overfitting and poor prediction of out-of-sample Ys).

Check penalized likelihood methods for model selection. (AIC, BIC, DIC...etc)

Model 1: High R and statistically significant model, however most of the predictors in this model are shown to be non-significant

Model 2: I removed the predictors that were not significant in Model 1, however the R decreased now but still significant model

Which one I should report? My aim is to identify the predictors associated with my dependent variable

Maybe you can alernatively consider the recursive least squares algorithm (RLS). RLS is the recursive application of the well-known least squares (LS) regression algorithm, so that each new data point is taken in account to modify (correct) a previous estimate of the parameters from some linear (or linearized) correlation thought to model the observed system. The method allows for the dynamical application of LS to time series acquired in real-time. As with LS, there may be several correlation equations with the corresponding set of dependent (observed) variables. For RLS with forgetting factor (RLS-FF), acquired data is weighted according to its age, with increased weight given to the most recent data.

Years ago, while investigating adaptive control and energetic optimization of aerobic fermenters, I have applied the RLS-FF algorithm to estimate the parameters from the KLa correlation, used to predict the O2 gas-liquid mass-transfer, hence giving increased weight to most recent data. Estimates were improved by imposing sinusoidal disturbance to air flow and agitation speed (manipulated variables). The power dissipated by agitation was accessed by a torque meter (pilot plant). The proposed (adaptive) control algorithm compared favourably with PID. Simulations assessed the effect of numerically generated white Gaussian noise (2-sigma truncated) and of first order delay. This investigation was reported at (MSc Thesis):

Thesis Controlo do Oxigénio Dissolvido em Fermentadores para Minimi...