The apt-cache search command will return all packages that have name in the package name or description:

# apt-cache search name

Once you have a package name, you can get more detailed information on the package using the apt-cache show and apt-cache showpkg commands.

# apt-cache show package_name
# apt-cache showpkg package_name
# apt-cache list package_name

Run apt list package, it will return the available version and if it is installed “[installed]”

# apt list package

# apt list bluez
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Listing... Done
bluez/stable,now 5.66-1 amd64 [installed]
------------------------------------------

# apt list --manual-installed=true

or

# apt-mark showmanual