Here I examine the various aspects of luck in board games.
Games with absolutely no luck are almost exclusively abstracts (see my post on Themes). All players know everything. Nothing is hidden. Nothing is random. Given sufficient computing power, the game could be "solved". Chess, Go, Yinsh and Medina fall into this category.
This includes games in which you do not see all of your opponent's information. Stratego, Lord of the Rings: The Confrontation, and block games including Wizard Kings fall into this category. You may have some information on how to proceed based on your opponent's moves, but in the end you must still guess the identity of the hidden units. Luck is part of your success. Deduction/induction games where you are trying to make successive guesses to find some answer also belong here. This includes games like Clue, Mastermind, Zendo, Sleuth, and Inkognito.
This is similar to hidden information. All players do something at the same time in secret. The results are then based on who chose what. There is usually some strategy based on the game situation and by knowing what choices the opponents are likely to make, but there is still some luck. Fairy Tale, 6 Nimmt!, and games with hidden auctions like Modern Art fall into this category.
In my opinion, this is the worst kind of luck. It is typically invoked by rolling dice or drawing cards. Where you end up is not by choice, making the game feel less like a game and more like an exercise. Monopoly is the quintessential roll-and-move game. Sometimes, the luck is mitigated by giving each player multiple pieces to choose from. Among games of this type are Sorry! and Verflixxt!.
Shared Unequal Luck
In some games, there is a random element that can potentially apply to any/all players, but affects them differently based on the position or other circumstances. In Settlers of Catan, a 2d6 roll determines what resources are generated, and by position, who gets them.
Shared Equal Luck
In other games. there is a random element that all players have an equal claim to. In Puerto Rico, the plantation tiles are random, but any player can claim one. Similarly, in Power Grid, the power plants are random, but all players can bid to purchase one. In Traders of Genoa, the starting location for the tower is random (usually), but all players can bid for control of its movement. In Goa, the setup of the board for the A and B stages of the game is random, but all players bid for the purchase of those tiles. In all these games, the only unequal aspect is player turn order (ie you may not be able to get what you want because another player gets to act before you do).
Luck of the Draw
Many games have this kind of element, usually in the form of cards or tiles. This kind of luck is typically mitigated by the fact that there are many draws over the course of the game. It comes down to how you use the resources you are getting. Carcassonne, Ingenious, Attika, Euphrates & Tigris, and China all have this element to varying degrees.
These games typically allow you to make any actions you want, but the results of those actions are resolved with a random element (eg dice or cards). Risk, Axis & Allies, and almost all wargames fit here.