Deciding to use an ANOVA model with interactions or testing significance without interaction I know this is a widely discussed topic and there are numerous sources and examples. However, most of the statistical jargon goes over my head. I was hoping that someone could simply explain my question.
So if one conducts an ANOVA test with interactions the following formula would be used: y~x*z
Without interaction: y~x+z
My questions are, in which circumstances would one use either test? Should both be used? Why does the result of dependent variables x and z produce different p-values with the two models and should this be discussed?   
 A: An interaction should be included in the model based on domain knowledge that would suggest the effect of x on y varies depending on z. If it's believed that x and z don't effect each other, then you would exclude the interaction term. When you are running a confirmatory study and your goal is inference, it's crucial to make these decisions before you see the data to prevent experimental bias. If you are conducting an exploratory study to generate hypotheses, you could build both models (y~x+z and y~x*z) and see which model better fits the data. Now that you've hypothesized a "better" model, the next step would be to collect a new sample of data and then test the "better" ANOVA on the new dataset to confirm your hypotheses.
The p-values are different because the two models are different. In y~x+z, the model is specified as $y = \beta_1 x_1 + \beta_2 x_2$ while y~x*z is $y = \beta_1 x_1 + \beta_2 x_2 + \beta_3 x_1 x_2$. Because the models have different parameters, the standard errors and parameter estimates will be different as the signal is attributed to different parts of the model. 
Some people may fit the interaction ANOVA y~x*z, see that the interaction term $\beta_3$ is insignificant (p > .05) and conclude that the interaction term doesn't need to be included because it's not significant. They'll resort to the normal ANOVA y~x+z and call it a day. This is called stepwise variable selection and is incorrect because using p-values to choose which parameters should be included in the model distorts inference and causes inflated error rates. See the TOPIC: Stepwise Variable Selection section of this discussion page for a more in depth explanation as to why you shouldn't choose variables to include in the model based on p-values. 
