'Direct Entry' means that their current ranking qualified them for a spot in the main draw of the tournament. Their name will sometimes appear in the draw prefixed with their ranking (and/or seed) number. Typically, the ranking taken into consideration is what the player was ranked 2 to 3 weeks before the tournament begins (when entries are finalized), so their ranking going into the tournament might be slightly different than their current ranking depending on if they played in a tournament between when entries were finalized and the tournament began.
'Wildcard' entry means that their ranking did not qualify them for a main draw position, and they were awarded one of the spots in the main draw reserved for a wild card. Their name will usually be prefixed with "WC" in this case. Wildcards are awarded by the tournament director typically and it is up to their discretion who to award them to. Also, the player would need to request a wildcard in order for it to be awarded to them. For a typical draw, the number of wildcards is not that great - usually between 2 to 4. For a major (128 player draw) - there are 8 wildcard spots.
'Qualifying' entry means that the player played 1 or more matches/rounds in a qualifying tournament and earned a spot in the main draw that was reserved as a qualifying spot. Their name will usually be prefixed with a "Q" in the draw in this case. The number of qualifying spots in the main draw usually depends on the size of the main draw - for a major, which has a main draw of 128 players, typically 16 spots will be reserved for qualifiers. Typically, players get chosen for qualifying rounds based on their current ranking - so those ranked high enough (but just outside the range for direct entry) get put into qualifying tournaments. Some tournaments also hold pre-qualifying rounds that are open to absolutely anyone eligible to play by virtue of being a pro and paying the entry fee. This is how the very beginning professionals (or maybe those coming back from an injury) work their way into tournaments.
The one other way a player can get into a main draw is by being what is called a "lucky loser" - which means the player that beat them to earn a spot in the main draw as a qualifier had to withdraw from the tournament before the 1st round began for some reason - so they were put into the main draw in place of that player.
Here is this year's Men's Singles draw from the Australian Open to take a look at if you want to apply what you just read: http://www.australianopen.com/en_AU/scores/draws/ms/msdraw.pdf