Each red or blue butterfly-shaped structure in the animation represents one chromosome that has replicated so that it's sister chromatids remain joined at their centromeres. When the chromosomes move together and pair up, they are called a homologous pair. The four chromatids together are called a tetrad and the process of pairing up is referred to as synapsis.
While paired, non-sister chromatids from opposite chromosomes touch and "cross over" each other. Each point of cross over is called a chiasma. While they are touching, the non-sister chromatids exchange genes. When the chromatids pull away later in meiosis, they are genetically different from when they started due to crossing over.
And because the chromatids are now genetically different, when they separate as chromosomes into haploid gametes (ie. sperm and egg), all the gametes will be genetically unique!
But don't forget there's another important process that increases the genetic variation of gametes, independent assortment.