Plastic snaps ("poppers"), like metal snaps come in a wide range of styles. Plastic snaps never rust and are completely hypo-allergenic.
Some plastic snaps are sold as a sew-on style found in most fabric stores, and can also be available as length of tape that has the snaps already attached (lunapads uses this). Some of these snap varieties can have "weak" snaps - meaning that they can come undone more easily under pressure than other styles. When used on something like a cloth pad, that is not under strain however, this is unlikely to cause any problems with pads coming un-snapped.
Polyacetal Resin snaps (often referred to as "industrial strength") are extremely durable, and are often used on cloth pads as well as baby products (such as modern cloth nappies/diapers). These need to be applied with a specialised tool or industrial press. These snaps have 3 components for each completed snap.
The "cap" (smooth top) has a "prong" in the centre, which goes through the fabric, and the snap press squashes that prong to flatten it against the "socket" or "stud" component which holds the 2 together to form one side of the snap. This process produces a strong snap that is less likely to pull out of the fabric than a metal snap. However the setup cost of the machine and snaps can be a disadvantage of this system.
While normally round, there are star, heart and other shaped plastic snaps available. The plastic snaps can come in a wide range of colours, including clear and glow in the dark.
Most often, snaps are seen holding the wings of a pad together, but some base+liner style pads use snaps to attach the liner to the base as well.
Some online stores offer a snap service, where you can send them your items to be snapped, and they will apply the snaps and send the item back.
Snap images above from http://www.verybaby.com