There are several options.
Basic alarm - neck pendant or wrist strap unit, with a base unit connected to the phone socket. Person presses the alarm on their pendant, which sends a radio signal to the base unit, which dials the alarm people. Alarm people will speak through the unit and if the person can speak, they can say what's happened. The units are good enough that they can hear and be heard through most of the house. If the person can't speak or can't be heard, the alarm people will contact the named contacts, or if they can't get them, will come out themselves. (This depends on the scheme - some of the private schemes won't go out themselves).
Other options - falls detector, again as a neck pendant or wrist unit. If it detects a sudden sharp movement, it triggers the base unit. These can be really useful for people who have seizures or are likely to fall and be unable to summon help. But, they're not much use for the people who flop down uncontrolledly into chairs as that can set them off.
There are lots of telecare options too - seizure detectors for under the mattress to monitor seizures at night - they can be set to just monitor number and frequency of seizures, or to get help, or to only get help if a seizure goes on for longer than a certain time. Pressure mats under the mattress can detect if someone has got out of bed and not got back into bed within a set time. Some systems are set so that if a person gets out of bed in the night, the lights come on lighting the way to the toilet and back, but leaving other areas in darkness.
Unburnt gas detectors will switch the cooker off at the mains if the gas is on but not lit. Heat detectors turn the cooker off if things start to burn, and/or call the fire brigade. Smoke detectors can alert the fire brigade as well. Flood detectors turn water off if they detect water on the floor from an overflowing sink or bath. You can get safety valve plugs which open to drain water if the weight of the water on top reaches a certain level which is an indication the sink or bath is too full. Door alarms will monitor if someone has gone out in the middle of the night, or you can get remote listening devices so that you can check the person at the door is a genuine caller and not a bogus workman/violent partner etc.
There are lots of telehealth developments too, although none of them are mainstream yet. Things like people with long term conditions doing their own daily readings - blood sugars, peak flows etc and transmitting the readings straight to their GP.