Elderly alarms are designed to be as simple as possible to set-up and use and are either mobile or landline-based. If a personal alarm is mobile it means there will be a SIM card (exactly like those used in phones) inside that allows the pendant to work wherever there is a mobile signal. This means that your relative can wear it in the garden or when they are out and about, safe in the knowledge that if they fall or get lost, they can press the button and get help.
Landline-based personal alarms are usually smaller, simpler to use, and cheaper, but will only work a certain distance from a ‘base unit’, plugged into the landline (i.e. within the house). When your relative presses the alarm button the monitoring team will only be able to talk to them through the base unit.
Both types of alarm are better at different things. Mobile alarms have more additional features, (sometimes) enable communication directly through the pendant, and work anywhere but landline alarms can be more reliable (as they almost always have signal), easier to use,  waterproof, and never require charging