I've heard that mobile phone cameras have lenses of inferior quality to those in DSLRs and point and shoot compact cameras. Is this true, and if so, why?
The lenses in mobile phones are typically made from molded plastic elements rather than ground glass elements, and these are in optically inferior to glass in general (higher dispersion, lower refractive indices). However molded elements have a big advantage in that they can be made into absolutely any shape, whereas glass is very difficult to grind into anything other than a sphere. Highly sophisticated designs thus are possible with every element being ashperical (high end DSLR lenses typically have only one or two aspherical elements).
Mobile phone camera lenses are in some ways better than DSLR lenses in that their resolution in terms of line pairs per millimeter is higher. But the sensors they are used with are so small that the total resolution across the image is lower. They also have to be faster (wider maximum aperture) to avoid diffraction, attempting to achieve the total image resolution of the best DSLR system is simply beyond the capabilities of the materials, given the size of phone cameras - which is the ultimate limiting factor.
The main difference between a DSLR and a mobile phone camera is not the lens, but the sensor. The sensor in a typical consumer DSLR is around about 24 x 16 mm, but that in a typical smartphone (the iPhone 6) is around 4.8 x 3.6 mm - or in other words, the DSLR sensor is around 22 times larger than that in a smartphone.
As Matt has answered, the lenses in a smartphone are also typically made of plastic rather than glass, but it doesn't really matter what they're made of - even if you put a super-engineered glass lens on a smartphone sensor, you still wouldn't get near the image quality of a large sensor camera.