It's quiet...
The Fog of War is a special condition where a dark shroud envelops the battlefield which prevents units from seeing, as well as attacking enemy units hidden within it, even if the location of the unit is otherwise clearly known. Statistical information on opposing armies is also hidden, such as funding, daily income, army value and unit count, but not opposing CO power bar charge.
Units push back the fog in a radius of tiles around them equal to their vision range. Owned properties will also provide vision of units on top of them, including hidden Stealths and Subs. Vision is collective and will be shared between units and across players in the case of allied teams as well.
Moving units into tiles beyond vision range carries a risk of being 'ambushed' or 'trapped' by a hidden enemy unit occupying any tile in the movement path. Upon encountering the hidden enemy unit, the ambushed unit's movement is immediately interrupted at the first point of contact, and the unit is unable to take any other actions for the remainder of the turn.
Some terrain tiles have additional interactions in fog - Woods and Reefs will always hide ground and sea units (but not air units) from vision, unless an allied unit is directly adjacent to them, while Mountains boost vision range of Infantry and Mechs that stand on top of them by 3, giving them as much vision range as a Recon. Rainy weather will reduce vision range for all units by 1 while it is falling, though it cannot reduce vision below 1.
A Blue Moon Infantry being trapped trying to enter an occupied forest.
Enabling fog of war changes the metagame of Advance Wars considerably - unlike in the default game which is essentially a game of perfect information, in fog, games are imperfect, where some information such as enemy army positions and compositions must be appropriately gathered before they can be known and others, such as a player's funding and property ownership, cannot be known for sure at all. This introduces strategic elements of risk management and prediction, where the opportunity cost of a move not must not only consider units that are visible, but also the potential risk of a previously unseen unit approaching from fog. On a tactical level, it also introduces information warfare - ensuring consistent vision as well as denying enemy vision can be a useful tool in skirmishes to both reveal favorable trades and allow more loose positioning requirements without retaliation.
The primary difference unit-wise is the addition of far more Recons in day-to-day fighting, having an excellent 5 vision while being cheap and fast to enable interrupting captures in the early game and transitioning to pure scouts in the later stages. Indirect units such as Artillery also benefit from fog, being more difficult to scout due to their range and able to steathily cover hotspot areas such as a contested property. These factors can be affected by map design - in the interests of producing faster gameplay, most are designed to be more open to prevent the risk factor from growing too high due to hidden indirects, with less chokepoints overall from mountain and sea and pipes tiles. These are the primary drivers behind the changes to the Standard Tier List brought on by fog.
... too quiet.
Sonja has an affinity for fog, with her day-to-day increasing vision range of all her units by 1, and bypasses the woods and reefs vision requirement when her powers are active. While not directly buffing her combat capabilities, Sonja nevertheless enjoys more tactical opportunities than other COs through these bonuses as she does not need to commit as many units (such as Recons) to obtaining vision, and in some cases gains attack vectors that would otherwise not be possible to exploit (such as attacking an enemy unit hidden in Woods with only one side accessible).