The only one of those categories that should get the "general" modifier is anesthetic - this is to contrast them with local anesthetics, which are the "numbing agents" you describe.
General anesthesics are agents that cause loss of consciousness. They prevent conscious perception of pain by causing loss of consciousness, rather than impacting pain pathways directly (some can also have direct analgesic effects, but this is not part of being a general anesthetic). It is still possible under general anesthesia for other systemic responses to pain to occur, and painful stimuli can cause arousal under lower levels of anesthesia. Therefore, general anesthesia is typically used alongside other methods for pain reduction.
There are several classes of general anesthetic. Commonly used are inhaled anesthetics like sevoflurane, intravenous anesthetics like propofol, and ketamine (which is a bit atypical). The mechanism by which general anesthetics causes loss of consciousness is a current area of research without clear answers, even though more is known about the molecular targets.
The drugs you call "numbing agents" are called local anesthetics. Local anesthetics block neuronal activity - they aren't particularly specific in a pharmacological sense. Instead, they gain specificity for pain by local administration.
The canonical class of local anesthetics are the '-caine's. Consumers are likely most familiar with lidocaine (available in over-the-counter creams for topical use) and procaine/Novocain (common in dentistry), but many other derivatives are used. Cocaine is also in this group, though typically used as a drug of abuse for other reasons, rather than as a local anesthetic.
The shared mechanism of these drugs is a block of sodium channels, which prevents neurotransmission. Local anesthetics can be applied topically, to target pain receptors, or can be injected in the vicinity of a nerve to block sensory neurotransmission through that nerve. Epidural anesthesia, such as used during child birth, is a particular use of local anesthesia targeted to the spinal cord.
Analgesics ("painkillers") more selectively target pain pathways. There are several classes, most medically relevant are the opioids, NSAIDs, and paracetamol/Tylenol. Opioids primarily target pain through inhibition of mu-opioid receptors; NSAIDs and related drugs through inhibition of COX-1/COX-2 and suppression of prostaglandin synthesis; the mechanism of paracetamol in pain relief is unknown.
Of course, all these drugs have affects on other systems, as well. Opioids in particular are not themselves typically considered general anesthetics (high doses can cause loss of consciousness, but the margins for safety are not acceptable for most purposes), but can aid in general anesthesia. The classification into these three categories is about the primary mechanisms of action at the doses they are used.