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In most games, 'your turn' means move a token, pawn, wathever around, but it also might include drawing a card, negotiate, whatever.
In most recent games, a move means more actions than just playing a piece (like within Chess or Go). More and more, a turn has several phases, some in which other players interact, some in which not.
Note that in many games moving your piece is no more or less than a delaying part of the game. Consider Cluedo, where you could jump from room to room, that would be no fun anymore. Here the moving (by using a dice) is purely meant as delaying factor, enabling players to get a slight advantage when they have some luck.
Take your turn in phases
As already set, many games have several phases during a players turn. Typical phasing looks like
A intruiging mechanism is limiting the number of actions (and movements) by some kind of budget (action points). Example: give each player 20 actions point / turn. Moving costs 5 pts, dealing costs 4 points, drawing a card 3 points, etc. As such each player can take several action, but is always limited to his budget. Typical examples are Tikal and Torres.
Moving your pieces
If some games, you have more than one representations, such a units, pawns, etc. An interesting issue is how these representation are moving. First question: if there is more than one represetantion, are they all moving the same (GO, Dames) or are there differences the way they move and/or the distance (Chess). Also, can this movement be influenced by you and/or your opponent(s), is it depending on luck or fixed (or a combination). Is the movement restricted to the gameboard (tiles) or is it free (pieces can move to any direction in any way the want e.g. on a table).
The (in-)famous snakes&ladders
A awfull lot of games use the well known 'snakes and ladders' mechanic to move your pawn forward. You roll a dice and move your pawn forward a number of fields, according the roll of the dice. Several fields contain special action, like you may jump forward/backward, loose a turn, take another turn, etc. The first one who gets to the last field wins. Some game companies did not even bother to change the lay out of the original game, but only added new graphics (note that the game mechanic is not copyrighted!). The main characteristic is that you move forward. As such you could call snakes/ladders a racing game, with obstacles. You just have to finish first.
As such this mechanic is often used in sponsored games. The example to the right a WWF gift, related to nature. It's the same boring game as snakes & ladders as I guess, as no additional mechanics were added. But then again, most of these games are for free. I have actually a number other sponsored games that use this mechanic, but hey, it's really boring. Seen one, seen 'm all.
Next to the snakes/ladders mechanic, where you must finish in the first place, many other games let you walk in circles, not to win by finishing first to obtain another objective (e.g. to win as much money as possible e.g. Monopoly).
As already mentioned, except the racing principle, nearly all game of this type have included events on certain field, where you can / must move forward fast, back ward, loose a turn etc. These are fixed events on fixed fields. Many of the (sponsored) games don't have any more mechanic than this one (mother goose, etc.)
adding money and things to buy
Some s&l alike games have added additional mechanics, such as buying 'insurrances' against disasters. Not just ending in first place before all the other players dor but e.g. aquiring the most of money or getting point for defeating monsters during the game may be the objective. A typical example is 1845, wich is an issurance game, where you get money every once in a while and where you can insure yourself against disasters. In this example for instance, as you pass 65 (you turn 65) you will get a retirement fee, if you have a pension scheme / fund bought some where during the game. Other games let you buy weapons etc. in order to help you fighting enemies (or opponents) you encounter. As a matter of fact, some of these items are one-time usable, some remain during the whole (unless they are being taken away from you). Another striking example might by monopoly (although there is no end, you keep on walking on the board, buying, trading and paying until you win/loose).
adding random event cards
Another typical add on to snakes & ladders is the use of cards (luck, chance, distaster, etc.). Instead of the fixed event on specific field, you draw a card on these fields. In some cases you draw a card from a specific deck, according to the collor or type of the field your land on. Some of these games require to draw an event card any turn.
moving backward and forward
Instead of just moving forward, you also may move backward (or sidewards if applicable). This gives you some kind of tactical advantage, since you might to decide to go the other way. A typical example is Trivial Pursuit.
having pawns with different characteristics
One could differentiate the pawns in a way that they react differtently to events, behave differently. Typically adventure / strategic games have this mechanic. Off course chess is another striking example where the different pawns have different characteristics. Should I mention that you overdo this mechanic. In some game the pawns have so many characteristics, that even can change during the game (AND SHOULD BE NOTED), that the game becomes unplayable. [note, the examples above are taken from games, where you can move in any direction] [see also: about pawns]
having pawns that can accumulate characteristics
Instead of drawing cards as simple events, pawns may gain permanent characteristics. [see also: about pawns]
Looking at the different aspects above, you would be easily tempted to re-invent Talisman (Games Workshop).
Any direction movements
In some of the most popular strategy games, you are allowed to move any way you want, left, right, up, down, diagonal, in some cases with restriction or obstacles. The most obvious example is chess. Other ones might by cluedo, dungeon quest, stratego, etc. Many of the characteristics as mentioned above go here as wel. Major difference is that you are 'free' to move.
Table top games moves
Mordheim and Warhammer are Games Workshop games (like many other table top games) where characters move by inches. A table or any surface can be the playground, usually enhenced by obstacles, buidlings and other scenery as shown. Building a complex scenery is a hobby on its own by the way.
Another issues is whether the movement you pieces can be modified either by the characteristics of your game-representation, you or your opponent. E.g. you randomize your movement by a dice, and have it changed by modifiers that come with the representation (e.g. a car drive 2D+3) or (Last speed +/- 1D). Keep in mind that if you choose to have different reprentations with different characteristics, each player has to know, or least must be able to verify, the speed you doing. Typical example is Warhammer and alike with lots of different representations and characteristics, that can be modified by adding additional gear.
The other thing you can ask yourself, what is the 'real' distance move. Does it move from tile to tile, from square to square, or from hex to hex? Or does it move actual distances (in inches / centimeters such as many wargames)?. Can it move to all sides (pawn v.s. queen in Chess)? Only straight (e.g. horse in Chess)? Can they skip over other pawns?
Diskwars has an interesting feature on moving the pieces. What you actually do is: you flip the disks (as they are called) over. So if the movement attribute is 3, you may flip the disk three times in any direction. In addition the disks come in various sizes (four?). It is a variation on real distances (like in Warhammer), but you actually don't need something to measure, you just flip the disks. It should be noted that flipping a lot will damage the pieces over time (the disk are made from hard-paper). There is an other interesting twist to the size of disk, the bigger they are, the easier to attack (with more pieces) (as more pieces can be put on larger disks, indicated a charge). Note that a large object does not neccesarely means it moves faster, a small disk can have an movement attribute of 10 (flipping it 10 times) while big one can have a movement attribute of 1.