Recirculation is the process of pulling the wort from the base of the mash tun or lauter tun and recirculating it back on to the top of the grain bed. Recirculation typically occurs after the end of the mashing process. As the hot wort is recirculated through the grain bed of the mash, the grains act as a particle filter clearing the wort. As the wort is recirculating, it becomes cleaner and less turbid until finally it is clear and ready to be passed to the boiling vessel.
A pump is typically used to recirculate the wort at a steady and controlled pace. In the case where a home brewer does not have a pump available, the wort may be drawn into a container and slowly poured back on top of the grain bed. The process can be repeated until the wort has become clear. Additionally, rice hulls may be added to a mash as a means of boosting the filtration capability of the grain bed.
A wide selection of home brewing recirculation pumps can be found here:
Secondary fermentation is the process of transferring your beer to a secondary fermentation vessel to allow the beer to complete its fermentation cycle and condition in a clean environment. The primary reason for a secondary fermentation is to improve the taste of a beer. Towards the end of the primary fermentation, much of the yeast, beer solids, and hop solids will fall out of the beer and form sediment on the bottom of the fermenter. If left in contact with the beer too long, the dead yeast and solids can impart off or undesirable flavors upon the beer. For this reason, many brewers choose to rack the beer off of the sediment into a secondary fermenter to allow the beer to finish out fermentation, clarify, and condition.
The need for secondary fermentation is somewhat dependent on the style and characteristics of the beer that you are creating. For instance, if I am brewing an American wheat hefeweizen, I probably will not go through the trouble of a secondary fermentation because it is a relatively low alcohol beer with a low flocculating strain of yeast. This means it will ferment quickly, so the beer is only in contact with the sediment for a short period of time, and much of the yeast will remain in suspension with this style of beer, so a yeasty taste and cloudy appearance is appropriate.
If I was brewing an IPA with a high starting gravity, and I wanted to highlight the hoppy flavor of the beer, I would certainly conduct a secondary fermentation to remove as much yeast and yeast flavor from the finished beer to help both with taste and clarity. Depending on the beer style, gravity, fermentation temperatures, yeast strain, and yeast health, a secondary fermentation can typically last anywhere from two weeks to several months. When conducting a secondary fermentation on certain beers, such as sours, the secondary fermentation can in some cases last over a year.
An imperial chocolate stout being racked into a secondary fermentation carboy,
Hot break is the clumping of proteins, solids, and tannins that fall out of the wort during the boil and eventually collect at the bottom of the kettle. A steady boil is the key to achieving a good hot break, which will typically occur 5-15 minutes after the boil begins. When the foam on top of your boil finally dissipates, you know that many of the proteins have coagulated and that your hot break has occurred. A hot break is important because it aids in removing undesirable and potentially off flavor causing tannins and compounds from the boil. It also helps improve clarity and reduces the risk of chill haze down the line.
Below is a photo of the hot break in a 20 gallon brew kettle, approximately 15 minutes after the boil first began.