Consular processing generally refers to the process of applying for an immigrant visa at a U.S. Embassy or Consulate abroad to obtain permanent residency (green card).   

The process involves many steps, which generally (but not always) begins with the filing of an I-130 visa petition with the appropriate office of US Citizenship & Immigration Services.  Once approved, the matter is forwarded to the National Visa Center for payment of fees and filing of various forms and documents for the immigrant visa application.  The National Visa Center then typically schedules a final immigrant visa interview at the US Embassy or Consulate abroad.  
The Department of State (USDOS) - and not USCIS - makes the final decision whether someone qualifies for an immigrant visa.  In assessing eligibility, the USDOS will evaluate whether the foreign national is admissible to the United States.  There are more than thirty grounds of inadmissibility that can bar someone from permanent residency.  Some of the more common grounds of inadmissibility are crimes, communicable diseases, helping someone enter the U.S. illegally, and public charge.  Some of the grounds of inadmissibility are waivable, while others are not.  It is important to consult with an experienced immigration attorney, such as the immigration attorneys at Wilson Law Group, to evaluate your particular situation, whether any of grounds of inadmissibiltiy apply, and if so whether a waiver of inadmissibility is available to you.