All tests currently available to test for the situation of being pregnant or not depend on testing for the presence of beta HCG or human chorionic gonadotropin in the blood or urine.
This hormone is only released by trophoblastic tissue, which usually is a growing foetus and its associated placenta or, rarely, from a choriocarcinoma or some other germ cell tumours.
False negative readings (from home kits) can result from the sensitivity of the test being lower than the current concentration of the HCG in the woman's urine. This can occur quite commonly before the 6th week of pregnancy (defined as 6 weeks from the last menstrual period) - beta HCG levels rise exponentially in the first two months or so of pregnancy so the earlier the test is performed, the higher the chance of a false negative result.
False positive results can result from weird things like choriocarcinomas. Truly rare.
Doctors can also perform quantitative serum beta HCG levels (as a glorified pregnancy test) to gauge the level of this stuff in the woman's blood.
This is commonly done in cases of suspected abortion, where a reading that is lower than the expected range of readings based on the patient's stated last menstrual period would be highly suggestive of a miscarriage.
Other uses of a quantitative beta HCG reading are in evaluation of ectopic pregnancy and in trophoblastic and other germ cell tumors.