The pavise was a pretty common feature of battlefields in the Later Middle Ages, at least in the Hundred Years War and the various Italian Wars of that time period (for German wars you'll have to ask elsewhere, I'm nowhere near qualified enough to answer for them).

The pavise has it's first firm reference in the 1260s in Italy (specifically in the army of Florence, while pavise is derived from Pavia it's not actually clear if the shield was for certain invented there or if it developed the connection later). It's probably safe to say that it was around before then, since the references don't seem to describe it as anything novel or new. We have a mention of pavises as well as the pavisarii who were responsible for deploying the shields and sometimes crouching behind them to ensure they stayed in place. They were closely linked to crossbowmen to the point of almost being considered a core piece of the crossbowman's equipment.

The Pavise appears in English records around 1356-60, where we have a record of over 500 pavises in the Tower of London (the Tower often functioned as an armoury during the middle ages). In 1365 we have a reference to a request from Charles V of France for good wood to be used in the construction of pavises. From this time period they became a piece of equipment popular with both sides of the Hundred Years War, with numbers of pavises in later references only increasing. In 1449 Philip the Good, Duke of Burgundy gave James II King of Scotland a massive wedding dowry which included 130 pavises (also guns, arms and armour, and lots of interesting stuff). Pavises seem to have remained popular at least to the end of the fifteenth century (we have more records of acquisitions in the middle of the century), but gradually disappeared after that.

As to what they were used for, crossbows were the start but the English certainly used them for longbows and references in the fifteenth century suggest that they were used in conjunction with early gunpowder weapons.

This is hardly a comprehensive history of the weapon, and if you'd like to read more I highly recommend reading Kelly DeVries' article "The introduction and use of the pavise in the Hundred Years War" which was published in Arms and Armour volume 4 (2007). It says everything I've included here and more, with great references for further research if you're interested.