Just adding on a clarification. The crossed structures the OP is talking about are actually sister chromatids linked by a centromere and they're attached lengthwise (| + |). Chromosomes are asymmetrical structures with a long arm and short arm and in the paired chromatid structure, the chromatids lie parallel to each other (short parallel to short and long parallel to long). They separate along the lengthwise axis.
Just to clarify, in a normal resting cell, chromosomes aren't found like this. These formations occur only during the preparation for cell division (be it mitosis or meiosis), during a particular phase called metaphase. In a resting (interphase) diploid cell, the chromosomes are not duplicated, and the cell is said to have 2n chromosomes, which is a full diploid complement. In the case of humans, n = 23. Half of those chromosomes came initially from the mother and half came from the father. In a resting diploid cell, the homologous (maternal and paternal) chromosomes are not spatially associated, and all the chromosomes are long, stringy and indistinct.
In the prelude to a mitotic division of a diploid cell, the chromosomes start condensing into fat cigar shaped structures that we're familiar with. In addition, DNA gets multiplied at this stage to give a total complement of 4n (tetraploidy). This is accomplished by each "cigar" forming a double, which are linked together to get the crossed structure we've been talking about. Each "cigar" in a crossed structure is properly called a sister chromatid. At the end of metaphase, these structures are fully formed and lined up along the mitotic spindle, ready to separate from one another. Note that in a mitotic division, the maternal and paternal DNA does *not* separate. Only the exactly duplicated sister chromatids separate. This results in 2 daughter cells, each with a perfect diploid (2n) complement of identical DNA (the two daughter cells are clones).
In meiosis things are somewhat more complicated. Meiosis happens only in gametogenic cells (germ cells), which have a 2n complement but must give rise to haploid (n) sperm cells or egg cells. A single sperm cell (or egg cell) contains only a maternal or paternal complement for a particular chromosome (but the choice is random). During the first meiotic division, the 2n germ cell will undergo duplication the usual way to give tetraploidy, *but* here the paternal and maternal chromosomes will become closely associated and crossing over events (chiasmata) often occur (adding to genetic variation). Pictorially, the associated maternal and paternal chromosomes look like 2 crosses with legs intertwined. Each of those cross-structures is one parental chromosome, comprising duplicated sister chromatids.
During the first meiotic division, the maternal and paternal chromosomes will separate to opposite ends to give a cross structure (2 chromatids to a cross) at each end. At this point maternal and paternal DNA have spatially separated except for the unpredictable cross-over events that happened before. Now the second meiotic division occurs without any further duplication. The sister chromatids at each end undergo a separation in a plane at right angles to the previous one to give haploid (n) cells. In total, a single germ cell gives 4 haploid gametes.
My explanation is long-winded, but I hope it answers all the OP's questions and more.