Traditional - Egypt

The game is played on an empty 5x5 board:

DROP PHASE - On each turn, each player drops two stones on some empty cells, except at the board center.
MOVE PHASE - One each turn, each player moves a stone. The player who dropped the last stones starts.
A stone moves to an adjacent (orthogonal) empty cell.
Stones can capture by custodian capture, i.e., an enemy stone is captured when two friendly stones are positioned one on either side of the enemy, either vertically of horizontally (but not diagonally).
If a stone moves to a position between two enemy stones it will not be captured.
After a capture, the player can move the capturing stone again, as long as it captures again.
A stone on the central square is safe from capture.
If a player cannot move, his opponent must move again and make an opening for him. 
GOAL - The winner is the player that capture all enemy stones, or the one with more stones on board once each player has built an impregnable barrier.

It is possible to create a barrier behind which there are only his own stones and these can be moved without fear of attack. The initial placing of the stones is important in planning such a barrier or in preventing the other player from making one.

Other board sizes are possible. A bigger board provides more strategic concerns.

An example

In this position, White has won, since he created a barrier that cannot be attacked, and he has more stones on board.

There is a ZRF to play Seega with Zillions, which is included in the Zillions original game set. Check this webpage.

This site presents another game with the same name. On a NxN board (N varies from 3 to 9) each player starts with N friendly stone on his first row. Each stone may move one or two cells in any direction (orthogonal and diagonal, it seems) but cannot jump over other stones. Wins the player that makes a orthogonal or diagonal N in-a-row (except, of course, in his first row). In my opinion, 7x7 seems already too big to play with these rules. In my trip to Egypt I saw in a regular shop, a 4x4 board with 4 stones for each player, which seems to fit this description.

A commercial game Seejah seems like a 7x7 variant of Seega, but I do not know the complete rules.