What is a multivariate random variable? I've been trying to read the Wikipedia article on multivariate random variables but I'm having trouble getting past the math. Is there a more intuitive explanation?
I'm assuming that a univariate random variable is the same as a random variable, for example: the outcome of throwing a die. Is the outcome of throwing two dice simultaneously a multivariate variable? 
 A: Yes, throwing two dice is a multivariate random variable. Specifically, you will get two independent and identically distributed discrete variables (assuming fair dice).
Throwing two dice and adding the results gets you a univariate random variable, with possible values between 2 and 12.
Picking a person at random and noting both their biological sex and their height is another multivariate random variable: a binary one and a more-or-less continuous one, and the two will not be independent any more.

One could argue that much of applied statistics is about drawing multivariate random variables and understanding just how exactly they are dependent. People will usually call all but one dimension "independent variables" and the last dimension "the dependent variable". You can also influence variables through your treatment.
A: With multivariate distributions, correlations between variables are important. If there is no correlation between to variable, then you basically have two univariate distributions. 
Throwing two die would be a multivariate distribution, but would probably have a correlation of zero (the exceptions to this are interesting. For example, if you have two loaded dice from the same factor, the two die would probably be correlated!).
In morphometrics people study how different measurements of animals vary. For example, one might care that weight and height are correlated. You might also appreciate this article's other biology and genomic examples
If simulations help you to learn by exploring data, here's some code to help you get started exploring a multivariate normal distribution in R:
    library(MASS)
Sigma <- matrix(c(10, 4, 10, 4),2,2) 
d <- mvrnorm(n = 3000, mu = c(10, 3), Sigma = Sigma)
plot(d)

That produces this figure:
Edit: I corrected my answer based upon the comment.
