In the big picture, the state recognizes that you still have a property interest in many things, such as your home, your company, your bank account, or your car. You still have ownership of most of the items that were yours in the first place, but the issue becomes exercising control over those items due to your being incarcerated. Someone who is incarcerated will usually grant their power of attorney to someone else to handle their affairs, including management of various property interests. That person has the authority to act in your stead, such as selling your car, managing your bank account, or renting out your house.
This can also include management of a business, however, most businesses will simply carry on without you, promoting the assistant manager to manager and carrying on as best they can - that's part of the point of businesses, is that they're simply intended to operate independent of the people who comprise it. Those businesses which cannot function without you will usually cease to operate if you are incarcerated. Remember, conviction of a crime is a long, long process, so it's not like you (or your business) wouldn't have had an opportunity to set up a succession plan or some sort.
Property that has an evidentiary value will be seized and held as long as needed. If you're arrested for a murder committed with a firearm and a gun is in your possession (or in your home, when they search it with a warrant), that gun is being held as evidence, and it will continue to be held as evidence until such time as it is no longer needed or until it is determined to not have evidentiary value (such as if analysis of the rifling indicates that this gun is not the murder weapon). That process may take a very long time. Evidentiary property can be reclaimed after it is no longer needed, though much property is not reclaimed and is as a result disposed of. I will note that this is how this should generally operate, but there are complexities to this regarding forfeiture which go outside of my scope of knowledge.
Property on your person when you're arrested and brought to jail will generally be processed and stored by the jail. This would include your clothes, your wallet, your car keys, etc. This will be searched and then stored, with the idea being that when you walk out of jail later (whether you were bonded out, served a sentence, were found innocent, etc.) that you'll have your property returned. This storage process often divides your "small property" from your clothes. "Small property" is your personal effects (wallet, keys, phone, jewelry, etc.) and valuables that can get sealed in a small plastic bag to verify that no one stole your watch or rings or whatever. Inmates in jail can then release their property to a family member, and this can be done either as "all property" (including the clothes, such as if you don't plan on getting out of jail) or just the "small property" (in case you want your wife to be able to get your car keys to get your truck from the bar you were at last night).
There may be additional institutional requirements regarding the storage of property, but that's a general guideline. When you are released to the street, your property will be returned to you. If you're being transferred to another facility (such as another jail or prison) your property will either be transferred with you or the facility may hold your property for a period of time to be picked up by someone you've designated (or you may be able to pay to have the property mailed). If the property is not picked up in that timeframe, it will be destroyed or forfeited. Such forfeited clothing is often added to a "lost and found" bin that may be used when inmates are released to the street without suitable clothing. (You might be surprised how many people get arrested in their birthday suit.)
The issue of personal property upon arrest does get more complicated when you're dealing with situations of people having more than just their clothes and wallet on them. Jails will often not store anything beyond "small property" and clothes due to space constraints, so if you are arrested with a backpack or a suitcase on hand, they will refuse to accept this property. Oftentimes, if you are arrested and have other people around, the arresting officer may let you give the backpack or suitcase to your friend so they simply don't have to deal with it. Assuming that this wasn't the case though, the way that this should generally go is that the arresting agency will hold the property for a short period of time before destroying it, and during that time, you can designate someone to come pick up your items. If your property is loose, rather than at least being organized into a backpack or a suitcase, such as if you're homeless and arrested from your camp... well, you're not supposed to be deprived of property without due process of law, but that's exactly what's going to happen.
As for other property not on your person, such as your belongings in your apartment, your car, or even your home, that starts to depend on the particular situation. If you have a loan or a lease, and you fail to pay your obligations on that loan or lease, then it will be confiscated, whether that means a repo man taking your truck or a landlord taking your apartment back over. All of those situations generally have well-worn civil processes, such as repossession of a vehicle for an unpaid loan or eviction from an apartment for unpaid rent, so I'll avoid going too deep into those possibilities. Again, you would usually have established someone as a power of attorney to deal with these things for you.