GARCH(1,1) is for a single time series. In GARCH(1,1) model, current volatility is influenced by past innovation to volatility.

Multivariate GARCH is model for two or more time series. In this case, current volatility of one time series is influenced not only by its own past innovation, but also by past innovations to volatilities of other time series.

Probably the best way how to start with multivariate models is to try to estimate such a model in some econometric software you have access to. For example, if you have access to stata, here is the manual:

http://www.stata.com/manuals13/tsmgarch.pdf

Particularly popular (due to its simplicity) is the Dynamic Conditional Correlation model of Engle:

http://pages.stern.nyu.edu/~rengle/dccfinal.pdf

A simple difference between the two is that, Garch(1,1) is used for modeling of univariate finacial time-series, that simultaneously model both mean and varience equation. But this garch model is unable to provide co-volitility thus, for that we have to go for Multivariate Garch model for modeling more than one financial series.

In multivariate GARCH also you need to model the AR-CH process (using implicitly the "Arrow of Time" Econophysics String Theoretic Dbranes Differentiable assumption which makes visual display on the computer meaningful - on this you can see some of my publications on my RG page) but in GARCH (1,1) you already have made a particular assumption in this regard, this besides anything else will affect the dimensionality of your solutions. Earl Chair Prof. Dr. SKM QC EPS Fellow (In) MES MRES MAICTE

I agree with Peter's answer, on the difference between the two, but also I advise you to see the papers by Engle and Kroner  (1995) VECH specification, and also Bollersleve and Engle BEKK models for multivariate GARCH specifications.

Would you please guide me. I have a model with 5 variables and want to see the relationships between them. I must use GARCH (1,1) or GARCH (2,2)?

Which one is multivariate and which one is uni-variate? 

You have to determine order p (autoregressive(AR)), and order q (moving average (MA)) for univariate GARCH (p , q) model if you are analyzing a single time series. Again, you have to determine order p and order q for multivariate GARCH model (p, q)  if you are analyzing two or more time series. By the way, you can check AR(p) and MA(q) by plotting ACF and PACF.

Let me add that Garch(1,1) can either be univariate, bivariate(one dependent variable and one independent variable), or multivariate(One dependent variable and more than one independent variable).

Thank you Eseosa Obadiaru

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