What is the result of meiosis?
In principle, Meiosis is only the process in which the haploid egg or sperm are generated. Have a look at this figure, which shows Meiosis I and II (from the Wikipedia):



During Meiosis I homologous recombination between homologous chromosomes can happen, the chromosomes are then distributed normally among the daughter cells. In Meiosis II the cells split again and here the sister chromatids are divided and the haploid gametes form. 

In the oocytes the process is in principle the same, but the cells get arrested in the so-called diplotene stage of Meiosis I (picture from here):



In the diplotene stage the chromatin is decondensated and transcriptionally active, this helps maintaining a correct chromosome structure, as the diplotene stage can last up to 50 years. When the cell receives the hormonal signal to enter the fallopian tube, Meiosis I is finished. The resulting cells are not divided symmetrical, building one big oocyte and one polar body containing the second set of chromosomes. Most vertebrate cells are arrested in this stage again until they are fertilized. Then another asymmetrical division is done, leading to a haploid cell with two polar bodies. This oocyte is then fertilized.

This process can happen almost at the same time in a lot of vertebrate species, including humans. This is probably what was meant by your textbook.

For further reading I can recommend: Meiosis and Fertilization from the NIH bookshelf.