While each law school differs in their specific approach to received applications, the following offers a general outline.
After submitting your application, law schools receive the following from LSAC:
The application itself
Your CAS Report
LSAT Writing (if you took the LSAT)
Letters of Recommendation you assigned to that particular school
All transcripts
(Rare, but if applicable): LSAC's Report on any Misconduct & Irregularity investigations/proceedings
Your application is processed and confirmed complete. An application may be incomplete for a variety of reasons, but most common are:
Missing Letters of Recommendation - contact your letter writers.
Processing issues - contact the admissions office and politely ask why your application is marked "complete."
Completed applications are transferred to first reviewers. This stage doesn't happen as soon as your application is received. Law schools typically process applications in batches/waves.*
Your file is read, notes are taken and your application is possibly assigned a score. Each school is different as to whether or not a scoring system is used. Multiple people may review, take notes and assign a score.
In many cases a Dean or Director of Admissions is the final reader in the process. Whether they read each application thoroughly or rely on notes of previous readers is at their discretion.
*It's important to remember that law schools do not review applications in the order received. Throughout the process there are criteria used to determine moving applications on to next steps/phases and what those next steps might be.
Spivey Consulting offers more detailed insight of the process in this article.