Mozzarella cheese comes in low-moisture and high-moisture varieties. Both can be frozen, but low-moisture cheese generally freezes better than the high-moisture type. Low-moisture mozzarella is typically shredded and used in dishes such as pizza, tacos or casseroles. High-moisture varieties generally are processed or fresh mozzarella that can be purchased in balls or bricks. Careful wrapping can help preserve the cheese in the freezer, though the texture may be different once it is thawed.
Shredded mozzarella can be frozen in the original package, or it can be removed and placed in airtight packaging, such as a sealed plastic bag. If you do not need to freeze mozzarella for very long, the original package can be put into a sealed bag before freezing. Cheese that will be frozen for up to two months might fare better in a more airtight package than the original one.
You can place the cheese in a sealing freezer bag and press the air out with your hands before closing it. Another method is to seal the bag except in one corner, and then roll it from the bottom, pushing air out as you go. A straw stuck in the open corner can also be used to carefully suck air out of the flattened bag in a sort of homemade vacuum seal. Removing as much air as possible can help you freeze mozzarella longer, and help maintain the original texture.
After you freeze mozzarella, the taste will probably not be different, but the texture may be drier and crumbly. How much it changes will most likely depend on the quality of the cheese. Additionally, some brands may not tolerate freezing as well as others, as they generally contain different amounts of moisture. Casseroles and pizzas are good options for using frozen mozzarella because the cheese will melt. Texture should not be much of an issue in dishes such as these.
Fresh or processed high-moisture mozzarella should probably be removed from the original packaging. This will allow you to wrap it in plastic wrap very tightly and keep air away from the surface better. After wrapping the cheese tightly and making sure it is completely covered, you may want to put that package inside a sealed freezer bag. Thin plastic bags that are not meant for freezing may not protect the cheese, so use one designed specifically for the freezer.
When you freeze mozzarella, the biggest problem you will usually face is dried out cheese. High-moisture cheese that should be smooth and creamy could end up slightly dry and crumbly after freezing. Careful wrapping and sealing should help reduce drying, but may not prevent it completely. If you want to get as close to the fresh texture as possible after thawing, you probably should not freeze mozzarella for more than one or two months.