Be in human form, be in the same level range and must have not killed the area boss

He is already being invaded or is in a place where you can not invade, such as Firelink Shrine (unless if you use the Cat Covenant Ring, since it will make you invade only Darkroot Garden)

• Using a Cracked Red Eye Orb
  You will spawn in as a Red Phantom a.k.a Dark Spirit or Black Phantom
  
  This is the method everyone is likely to know. All you need to do is: find an area where you can actually press "Use item" on it, Be human (That means that you have done "Reverse Hollowing" with one humanity at a bonfire)
  
  
  
• Using the Red Eye Orb
  You will spawn in as a Red Phantom a.k.a Dark Spirit or Black Phantom
  
  This item is identical to "Cracked Red Eye Orb" with the two differences being: You have to be in the Darkwraith covenant and you the item is not consumed upon use.
  
  
  
• Using the Red Sign Soapstone
  You will spawn in as a Red Phantom a.k.a Dark Spirit or Black Phantom
  
  In order to invade with this item, you have to place it down like you would with a White Sign Soapstone (Co-op summon sign), except that once a host summons you, you will spawn in as an invader where you placed the sign. The requirements to invade with this are the same as with the Cracked Red Eye Orb, except that the summon level range is different, and the item is not consumed upon use.
  
  
  
• Using the Blue Eye Orb
  You will spawn in as a Blue Phantom a.k.a Spirit of Vengeance
  
  To use this item you do not have to be in human form, but you do need to be in the Darkmoon Blade covenant, the item is not consumed upon use, just like the Red Eye Orb.
  In order to invade with this item, the host must be in human form, must have 1 sin or more (Sin = Indictments), must be in correct level range and must have not killed the boss of the area.
  Each time you kill a host after invading with this item, they lose 1 sin.
  
  
  
• Using the Cat Covenant Ring
  You will spawn in as a Blue Phantom a.k.a Spirit of Vengeance
  
  In order to invade with this item, you have to be in the Forest Hunter covenant, and you have to equip it, once equipped, you will invade an eligible host connected to your node. You do not need to be in human form to use this item.
  To invade, the host must be in human form, in Darkroot Garden, must not be in the Forest Hunter covenant and must have not killed the area boss (Sif). Although you will invade as a Spirit of Vengeance, the host does not need to have any sin, they only must be in Darkroot Garden. You can equip this ring almost anywhere and you can still invade people in Darkroot Garden.
  
  
  
• Interacting with a Gravelord Soul Sign
  You will spawn in as a Blue Phantom a.k.a Spirit of Vengeance
  
  If you see one of these signs, (which have a very distinct dark style to them) it means that you have been "Gravelorded", an eligible host in the Gravelord Servant covenant has used an "Eye of Death" to make tough enemies spawn in your world, the tough enemies look like Red Phantoms and do a lot more damage. When you interact with the sign, you will invade the world of the Gravelord Servant and once you kill him, the Red Phantom enemies will stop appearing from your world. You do not have to be in human form to invade with this method.
  
  
  
• Using the Darkmoon Blade Covenant Ring
  You will spawn in as a Blue Phantom a.k.a Spirit of Vengeance
  
  Something unique to this one is that the host will not get the "<Player> has invaded!" popup.
  You will have to be in the Blade of the Darkmoon covenant and must have the specific ring equipped, you must be in Anor Londo. You do not have to be in human form as far as I know.
  
  
  
• Using the Dragon Eye
  You will spawn in as a Dragon Spirit
  
  In order to use this item, you must be a member of the Path of the Dragon covenant, you do not need to be in human form.
  Upon using this item, you will place down a summon sign that resembles a dragon eye, the host can only see it and summon you if they have not defeated the area boss, are in human form and do not have any phantoms present in their world. Upon being summoned by this sign, you will not spawn where you placed the sign, oddly enough.

You must be in human form and you must have not defeated the area boss.


Just reverse-hollow yourself and wait in a place where you have not killed the area boss, great example are Darkroot Garden. If you are not a Forest Hunter, Forest Hunter covenant members will try to invade you as long as you are in Darkroot Garden, even if you are a Forest Hunter, but still human, you are likely to be invaded by a Red Eye Orb.
You can also go to another very popular place, Oolacile Township. You are guaranteed to find PVP action as long as you are human and are connected to some people. You will often see Red Sign Soapstones laying around, interact with one and it will summon a Red Phantom to fight you.

Although not a very good method always, it might work. You must be in the Gravelord Servant covenant, and in human form.
In this method, you go to a place where you can be invaded and use an "Eye of Death" to spawn Gravelord Phantoms in the worlds of connected players, this will also make the Gravelord Soul Sign appear in their worlds, once they interact with it, they will invade you.

Wulf's Dark Souls Connectivity Mod a.k.a DSCM http://wulf2k.ca/PC/DaS/DSCM.html This enables you to manually connect to your friends and with "DSCM-net" it makes you connect to other players much faster, it also displays information of who you are connected to.
DsFix(for 30fps cap unlock) http://blog.metaclassofnil.com/?p=684&cpage=1#comment-281676
Dark Souls PVP Watchdog a.k.a DSPW (anticheat) This one checks whether the invader or host is cheating by editing their stats or soul level, if they are, you can safely disconnect from their world or kick them out of yours.

White Phantoms and Gold Phantoms are summoned by White Sign Soapstones for co-op, they can engage in combat with enemies and any invaders.

Red Phantom (Dark Spirit)
Either summoned via Red Sign Soapstone, or invades with Red Eye Orb or Cracked Red Eye Orb. Red Phantoms Can hurt friendly phantoms, other invader phantoms and even other Red Phantoms.

Blue Phantom (Spirit of Vengeance)
Invades with Blue Eye Orb, Darkmoon Blade Ring, Gravelord Soul Sign or Cat Covenant Ring.
These kind of phantoms can engage in combat with friendly phantoms and Red Phantoms but not other Blue Phantoms

Dragon Spirit
Invades by placing down a Dragon Eye sign. I do not know how this kind of phantom behaves against other phantoms, since it can only invade a lone host.

X = Soul Level of Invader
Y = Soul Level of Host

Red Sign Soapstone and Cat Covenant Ring
Lower than host SL: X*0.9-10
Higher than host SL: X<Y
So a SL 100 player can invade a player between SL 80-713

Cracked/Red Eye Orb
Lower than host SL: X*0.9
Higher than host SL: X<Y
So a SL 100 player can invade a player between SL 90-713

Eye of Death and Dragon Eye
Lower than host SL: X*0.9-10
Higher than host SL: X*1.1+10
So a SL 100 player can invade a player between SL 80-120

Blue Eye Orb and Darkmoon Blade Ring
Lower than host SL: X - (50+0.2*X)
Higher than host SL: X*1.1+10
So a SL 100 player can invade a player between SL 30-120

If you want to check quickly, use this site: https://mpql.net/tools/dark-souls/

Feel free to contact me if I missed something or some of the data is invalid