To give teams a fair shot at free agents and to prevent abuse of pickup and drop functionality, Sports.ws implements a waivers system. All players that are dropped are placed on waivers immediately. There they will remain for your league's waiver wait period (usually 36 hours). At the end of the period, if a team has expressed interest in a player, they will automatically pick them up. If more than one has expressed interest, the team with the worst record (at time of expressing interest) gets them.
To express interest in players, or to peruse the players on waivers, click the LINEUP tab.

Scroll down to the very bottom of the page. If there are any players on waivers, you should see something like this:

Above you see four players on waivers, all with different times of availibility. You can express interest in a player as long as the number of players on your roster plus the number of interests express does not exceed 13. This means you usually have a full roster of 12 and can express interest in one player.
To express interest, simply click the "Express Interest" button next to the player of choice. When you express interest, nothing happens except your team gets put in the queue of teams waiting to get the player. You'll get put ahead of any team with a better record, and behind any time with a worse record. If two teams have the same record, the team that expressed interest first gets priority.
After expressing interest in a player, you will see something similar to:

In this example, you're first in line because of your record or timing. You will not know which, nor how many teams are behind you. If you drop interest, then express again, a new priority will be calculated. In this case, when availibility time is reached, if your rank is still 1, you'd pick up the player automatically.
You can see the pickup on your trade page by clicking the TRADE tab in your league home.

A typical waivers pickup logged looks like:

There might also come a time when you express interest in a player and are immediately pushed back in line because teams with worse records already have expressed interest. Your waivers list might look like:

You can leave your interest and hope that the three teams in front drop theirs (usually they do this if they find a free agent and need roster space, or if they find another player they want on waivers), or you can drop interest by clicking the "Drop Interest" button. There is no penalty for expressing interest in a player and not getting him.
Also, if you have enough room (less than 12 players on your roster), you can express interest in multiple players at once. Your list would look like:

The same waiver rules apply to each individually, and the total number of players on your roster and interests expressed has to stay at 13 or below.
In the best, most competitive leagues, waivers help balance things out well. While it is important to reward owners that are on top of things and quick to make moves, many prefer this system because they can't always be in front of a computer or track the trade log as regularly as they would like. Also, the teams at the bottom of the league have a prolonged interest in the league when they know they have priority with waivers.