How to test for differences in ratings for two devices where each device is rated on a multiple item scale? I have two devices, and I'm trying to figure out which one of them is preferred by users.
Thirty people have rated each device on a 12-item questionnaire which asks them to grade the device on 12 parameters.


*

*What statistical tool should I use to decide which device is better? (just comparing the grades is not enough)

*Can I test for correlation between the "device type" and some kind of weighted grade?

 A: First, I encourage you to say more about why "just comparing the grades is not enough."  But more to your point, although the use of correlation is not out of the question, you and the people to whom you are reporting would no doubt be better off viewing the problem as one of group differences.  Then your question becomes "Is Device A, on average, viewed more or less favorably than Device B?" For this, T-tests are the usual starting place. 
Or perhaps you have highly non-normal distributions, in which case means are not the best way to represent the typical score.  In such a situation, you would want to test for differences using medians, or ranks.  So you would want to get familiar with the techniques of Median tests and/or Mann-Whitney tests involving ranks.  
(Since the same group of 30 people have supplied each set of ratings, you have what are called "dependent sets of scores" and may need to look for a more specialized test that takes this into account.  Others on this site will probably have ideas.)
Another approach would be to average the 12 ratings, creating a single summary score for each person with regard to each device.  This average would probably be normally distributed, and so you could use a single Dependent-Means T-test of differences.  Just to make things confusing, this is also called a Correlated Means test.  But if you are going to create such a summary score, you might want to check and see that it is an effective one, i.e., that all 12 component scores are correlated enough to "play their part" in forming the summary score.  this is a reliability problem, so you could look into that as well.  and one measure of reliability that would help you would be Cronbach's alpha.
A: On inference
If you insist on inferential statistics, use a Hotelling $T^2$ test to get an aggregate measure. This is the multivariate equivalent of Student's $t$ test; it will test for differences in multiple question results simultaneously. If you want more detailed statistical tests, you can follow this test with what @rolando2 suggested.
But since your sample is small and you don't seem to be trying to develop some grand, basic model about how the universe (or at least people) work, I doubt that you need inferential statistics.
Approach
I'd start the analysis with a box plot of the differences in ratings for each of the 12 questions. You should also look at the ratings for the individual questionnaires rather than the differences.
differences<-round(2*as.data.frame(matrix(rnorm(30*12),12,12)))
boxplot(differences)
dev.off()


If you don't see a clear difference on one question, there probably isn't a large difference on that question. If you do see a difference on one question, you can think about why they differed on that question.
You don't have enough degrees of freedom to run the $T^2$ test on all of the questions, but you can run it on a subset.
T2.test(differences[1:4])

