Soon after Curie brothers’ discovery of piezoelectricity in crystals like quartz, cane sugar, and Rochelle salt in 1880, mathematician Gabriel Lippman predicted the existence of a reverse piezoelectric effect through his calculations. He hypothesized that when an electric field is applied to these crystals, they will change their shape, lengthening or shortening according to the polarity of the field. The Curie brothers later confirmed this experimentally. You can read more about the history and specifics of the discovery here.

The initial use of piezoelectricity in sonar in World War I created intense international interest in piezoelectric devices. World War II and the ensuing decades saw the birth of a variety of synthetic piezoelectric materials. They were processed to performance better than their natural counterparts in terms of their piezoelectric constants, ease of manufacturability and better stability with regards to temperature and humidity. Barium titanate and Lead Zirconate Titanate (PZT) are examples of synthetic piezoelectric materials used today.

Barium Titanate's unit cell houses a permanent dipole and a voltage even without any applied mechanical stress or an electric field. This is due to the unique non-symmetric positioning of Barium, Titanium and Oxygen ions in the tetragonally shaped unit cell. Each titanium ion is surrounded by an octahedral arrangement of oxygen ions. But the titanium is slightly displaced from the very center of the octahedron and this charge separation gives rise to a permanent dipole. However, different groups of dipoles are usually randomly oriented across the material and cancel each other out. To extract useful piezoelectricity, materials like Barium Titanate are subjected to additional processing called poling to re-orient the dipoles. After this stage, a mechanical force applied in an appropriate direction can increase the existing voltage by thousands of volts. When you do the reverse, i.e., apply an electric field to the crystal, there is movement of ions due to their electrical interactions with the field causing the crystal to lengthen or shorten. 

Piezoelectric materials and devices have come a long way since their initial use in sonar technology. For example, in watches, electricity from the watch battery causes the piezo crystal to oscillate thousands of times per second, and circuits in the watch convert these oscillations to a once-per-second digital beat. Look here if you want to know more about how quartz watches operate. Just a flick of a switch can trigger a piezo crystal inside a BBQ grill lighter and generate thousands of volts, creating a spark that ignites the gas. Because very high voltages correspond to only tiny changes in the length of the piezo crystal, they are excellent in applications that require very small amounts of movement with micrometer precision, like lens adjustments in microscopes. If you want to know more about applications of piezoelectric materials, read this article.

You can read about some exciting energy harvesting projects like sustainable dance floors that use piezoelectricity here, here and here.

This link will give you a good idea of the kind of quantitative calculations that go into modeling and building structures that employ piezoelectricity.