Estrogens are the hormones produced by ovaries. A very low level of estrogen is usually found in a case of non-functioning ovaries (in cases of ovarian failure, menopause, etc.). The consequences of having a very low level of estrogen are the same as during menopause (vaginal dryness, decreased libido, etc.).
In a woman's normal menstrual cycle, for two or three days there is a low level of oestrogen (the first few days of menstruation) but the oestrogens quickly start to rise, until they reach a maximum, when ovulation occurs.