How do the VNAV and LNAV work on the 737NG?
I see some confusion about the Autopilot system here. First let's clear something up:


The B737 has 3 guidance: speed, horizontal and vertical. These are displayed on the top of the PFD from left to right.
Engaging Flight Director displays the guidance but not to follow it, engaging the Autopilot tells the plane to follow the guidance.
Each guidance has several modes. E.g. Possible horizontal guidance modes include "HDG SEL", "HDG HLD", "LNAV" etc. Possible vertical guidance modes include "V/S", "ALT HLD", "VNAV" etc.


LNAV is just one of the horizontal guidance modes. The FMC decides what is LNAV. If LNAV is not engaged, then the FMC does not take part in horizontal navigation.

Your confusion likely arises from armed modes. The B737 allows you to select one active mode, plus one armed mode. For example, consider the LNAV path in FMC is heading North (360 degrees). The plane is to the East of the intended path (right side). By doing the following:


Set the heading window to "340".
Set "HDL SEL" as active mode.
Set "LNAV" as armed mode.


The plane is on a heading to join the LNAV course. When the plane is close*1, the horizontal mode will automatically change from "HDG SEL" to "LNAV".

Similarly, "V/S" can be used to capture VNAV profile when the plane is above or below the intended altitude.

*1: The plane has to be within a certain nautical miles and the intercept angle cannot exceed a certain degree. I will update the exact number when I get reference to the B737 manuals.