Mini IVF is also known as "minimal stimulation IVF." The ovaries of a woman are minimally stimulated, using low doses of gonadotropins, follicle-stimulating hormone and luteinizing hormone, in order to encourage the growth of a small cohort of eggs. Once mature, the eggs are retrieved surgically and fertilized in the lab, and the resulting embryos are transferred back into the woman's uterus. The process is identical to regular IVF, except that the goal is to create only a few embryos for transfer.
There are both advantages and disadvantages to using this process. Mini IVF is less expensive, primarily because you are using much less medication. Most clinics offer a discounted price as well. However, keep in mind that because you are producing fewer eggs (and embryos), it is significantly less likely that you will have excess embryos to freeze at the end of the cycle. This may not seem important if you only want one child, but if you don't become pregnant, multiple cycles of mini IVF could easily be more expensive then one cycle of regular IVF.
Before deciding, think about why you're considering mini IVF -- are you nervous about the cost, or does taking less medication appeal to you? Do you only want one child and don't want the ethical concern about extra embryos? Mini IVF is not appropriate for everyone; you should have a candid conversation with your partner and physician before proceeding.