Trap Monsters are a type of Continuous Trap Card that is Special Summoned as a Monster Card after activation, being treated as both a monster and a Trap Card simultaneously. In the manga, they are a separate type of Trap Card.
Trap Monsters are Special Summoned to the field from the Spell & Trap Zone after they are activated, and are subsequently treated as both Monster Cards and Trap Cards. As such, they take up both a Monster Zone and a Spell & Trap Zone at the same time.
If they are negated with "Jinzo" or "Royal Decree", they will return to the Spell & Trap Zone as Continuous Trap Cards with no effect, and will remain there even after "Jinzo" or "Royal Decree" are removed. If "Book of Moon" is used against an already active Trap Monster, the Trap Monster is set as a Trap Card again and can be activated and Special Summoned the next turn.
If a card or effect that destroys monsters specifically (like "Mirror Force") is activated, cards like "My Body as a Shield" can be activated to protect a Trap Monster, whereas cards like "Fake Trap" cannot be; likewise, if a card or effect that destroys Trap cards specifically (like "Heavy Storm") is activated, cards like "Fake Trap" can be activated, whereas cards like "My Body as a Shield" cannot. If a card or effect which can destroy both monsters and Traps (like "Raigeki Break") is activated, both cards like "My Body as a Shield" and "Fake Trap" can be activated. In all cases, effects like that of "Imperial Custom" prevent Trap Monsters from being destroyed, even by battle.
If a Trap Monster is treated as an Equip Card from the Monster Zone, such as due to the effect of "Relinquished", it is treated only as an Equip Spell Card, not a Trap Card or Monster Card.
Trap Monsters are typically used as an emergency defense to stonewall an opponent ("Metal Reflect Slime" is particularly useful in this regard). They can also used to provide quick Tribute Fodder ("Metal Reflect Slime" is great when combined with "Level Eater"), such as the Monarchs, and can in some cases be used offensively as well. Also, because they don't count as monsters while in a player's Deck or Graveyard, some effects might work on them as a Trap Card, while others (like the effect of "Morphing Jar #2") don't see them as a monster. This can be helpful in some cases (see All-Out Attacks OTK).
Another use for Trap Monsters is for Synchro Summoning or Xyz Summoning. Since most Trap Monsters are Level 4, they can be used to quickly Synchro Summon a Level 5-8 monster or Xyz Summon a Rank 4 monster. "Metal Reflect Slime", which is too high a level to be useful for a Synchro Summon, can be used for Rank 10 Monsters such as "Superdreadnought Rail Cannon Gustav Max". If combined with a Level 1 Tuner monster, they can be also used to Synchro Summon "Star Eater". "Shape Sister" is a Level 2 Tuner Trap Monster that can be used to assist in Synchro Summoning.
"Imperial Custom" makes Continuous Trap Cards indestructible. If you control a copy of "Imperial Custom" and a Trap Monster, the Trap Monster will not be able to be destroyed by battle or by a card effect. This combination is especially potent in defensive decks.
Trap Monsters are also useful in a Uria Deck, as they provide the Deck with monsters that can also be used as a Tributes to Summon "Uria, Lord of Searing Flames" as well as give him extra ATK points.
It is usually a good idea to have "Trap Eater" and "Double Snare" and "Splendid Venus" in the Side Deck of a Deck using high concentrations of these cards, so that you can remove cards such as "Royal Decree" and "Jinzo" which obstruct the use of Trap Monsters. "Magic Planter" is also useful, allowing the player to send Trap Monsters to the Graveyard to draw cards; this is particularly useful if the Trap Monsters have their effects negated, which causes them to remain on the field as Continuous Trap Cards only.
"Judgment of Anubis" is useful to protect Trap Monsters from "Mystical Space Typhoon".