A mole can come back after removal, usually if some mole tissue is left in place during the mole removal procedure. This can occur for various reasons, such as insufficient treatment depth or duration. The most effective way of removing a mole so that it doesn't return is by surgical excision through the full skin thickness, including a margin of normal skin all the way around the removed mole.
A mole or skin lesion can come back after removal if some mole tissue is left behind at the time of the initial removal. The best way of avoiding this is to have the entire mole or skin lesion removed by cutting through the full thickness of the skin, with a margin of clear skin (usually at least 2mm) all around the mole. This is the procedure used when a doctor suspects that the mole might in fact be a melanoma. The excised skin can be sent to a laboratory for a skin pathologist to determine the diagnosis and confirm that the entire mole or skin lesion has been removed. If complete removal is confirmed, it's extremely unlikely that the mole would ever reappear. The disadvantage with this method is that it tends to cause worse scarring than other methods.
Moles are more likely to return after treatment with cryotherapy (freezing) because often the lesion is not frozen long enough for all of the cells to be destroyed. Freezing for a long time is painful and can leave a scar, so many doctors attempt a "gentle" treatment of the lesion, especially if it is relatively low risk like a solar keratosis or harmless like a seborrhoeic keratosis.
If a mole returns after treatment, it can usually be re-treated successfully. However, the initial removal process and subsequent healing can significantly change the appearance of the mole when it grows back, especially in the case of getting black or dark moles removed — returning moles sometimes resemble melanoma. If a second doctor examines the returning mole and there is no original histopathology result available, they may decide that the mole needs to be completely excised.