What happens if I use Z-test instead of T test? I know that when sample size goes large, they basically give the same result.
My question is that 
what happens if I use z-test instead of t-test when sample size is not large. 
I guess the type I error increases and power increases. Am I correct? Thank you.
 A: Because the critical values are smaller with the Z, the rejection region is larger.

So yes, significance level goes up, and consequently the rest of the power curve comes with it, as it does whenever you move the significance level.

A: There is little to add to @Glen_b expert and eloquent discussion; rather, there is only room for making concepts less precise with the excuse or intention of appealing to intuition. That being said, here're a couple of plots that helped me come to terms with these concepts:
1. t-Student Distributions have "fatter" tails:
These tend towards the normal distribution as the sample size (or degrees of freedom) increase. Consequently, there are more points lying in the asymptotes, and to determine a certain risk alpha, the cut-off point of the test statistic would have to be slid towards the right (let's leave aside two-tail tests). The comparison is thus:

2. The farther away the cut-off, the lower the power:
So we slide the cutoff value to the right, and in doing so, we stay within the NULL a longer stretch before we reject it. Looking at it from the alternative there is a larger area of its corresponding curve sub-tending to the left of the cut-off value (beta), and a smaller slice to the right (the power). Just like this:

Notice the funny looking t-distribution under the alternative, which is a tentative approximation to a non-central t with a delta parameter of $2$. Under the NULL the distribution is central with $2\,df$. The cut-off value for a one-sided risk alpha of $0.5\,\%$ is shown as vertical straight lines on both the Z-test (above in blue) and the t-test (below in red).
