MSI is often counterintuitive and somewhat complicated under the hood. However, to over-simplify an MSI file contains one or more "Features" - and these features collectively constitute the "bits of the application" as you put it.
You can generally find a list of these features by running the setup interactively, and navigate to the customize install dialog (not always present). Features that show up here are normally the "user configurable" parts of the application that can be chosen for exclusion or inclusion. Typical features are: Dictionaries, Samples, Plug-Ins, etc... Anything that the application doesn't really need to function, but that's included in the setup.
There are generally 2 ways to customize an MSI installation: using msiexec.exe custom command lines, or using transform files.
1: msiexec.exe command line:
The simplest and light-weight way of controlling what features are installed during an installation, is to specify your feature selection using the msiexec.exe command line. There is a whole family of properties used for feature configuration. But, in most cases it is sufficient to specify ADDLOCAL:
msiexec.exe /i myinstaller.msi ADDLOCAL="Program,Dictionaries" /qn
The above command line specifies that the features "Program" and "Dictionaries" should be installed locally (feature names are cases-sensitive!). This is generally enough, but you can also specify any features that you want to remove using the REMOVE property in a similar fashion. A special switch is ADDLOCAL=ALL which will install all features in the MSI on the local disk (provided there is not additional logic in the MSI to override this). ADDLOCAL property on MSDN:
2: Transforms
MSI files are essentially SQL-databases wrapped in COM structured storage files. Transform files are "partial databases" constructed via installation tools such as Orca, Installshield or Wise, etc... These transforms can customize or override almost all settings or database fields in an MSI - including what "parts of the application" (features) are installed. After creating a transform you specify its application to the MSI at the msiexec.exe command line:
msiexec.exe /i myinstaller.msi TRANSFORMS="mytransform.mst" /qn
Windows Installer will then merge the MSI and the tranform before installation starts. This is the approach used by large organizations who want full control of how the MSI gets installed. TRANSFORMS property on MSDN
A reasonably good: summary of Windows Installer.