The designations of Navy ships is arcane, but very precise.
LPH is (amphibious) Landing (ship) (for) Personnel (and with) Helicopter (deck). This is also known as "Landing-Platform-Helo" as a mnemonic from the initials; the 'official' (and virtually unused) title is "Helicopter Landing ship."
LHA is (amphibious) Landing (ship) (with) Helicopter (deck) (shore support ) Assault. this is also known as "Amphibious Attack Helicopter Carrier."
LPD is (amphibious) Landing (ship) (for) Personnel (with) (well) Deck. This is also known as a "Landing-Platform-Dock" from its initials. IIRC, the official title is Amphibious Assault ship. There is a related and similar ship, the LHD, which has a small helo-landing deck at the stern.
The main feature is a floodable well deck (near universally mis-called a "dock' or "dry dock") from which landing craft can be launched (and one LCAC; Landing Craft, Air Cushion) can be stowed internally.
LSD is (amphibious) Landing Ship (with) (well) Deck. These are also known as "Landing-Ship-Dock" from the intials, again with the misnomer of "dock' for "well deck." LSD can devote as much as 2/3 to 1/2 their overall length to the well deck, to the purpose of carrying a large number of relatively-fragile landing craft (often nested one within the other). The pone top of the well deck allows the use of cranes to load the landing craft while stable and dry, and let them leave 'wet' afterwards. Since the 60's all LSD have carried some sort of helo deck; the more modern classed some hangaring provision for helos is also included.
LST was (amphibious) Landing Ship (with) Tank (deck). The term "Landing-Ship,Tank" is too often used--a hold-over from British RN usage, and not at all related to armored fighting vehicles. LST are the largest vessels designed to "strike ground." Which is actually quite complicated. Further, "doors" in the bows are one of the least-desirable design constraints in nautical architecture. (very much a pain to navigate on open water with a cross-wind.)
In WWII, there were a number of Auxiliaries used. AP just meant Auxiliary (Ship) Personnel (carrying) and was most used for militarized Liners. AK encompassed merchant cargo vessels in military service. With the advent of Liberty and Victory ship construction, a way was needed to to designate those hulls modified for specific Amphibious use, thus the suffixed "A" creating APA from AP; AKA from AK. As Amphibious tactics gelled, it was felt that the APA & AKA ought carry an "L" as their companion vessels did. This gave us the LPA and LKA.
There was almost no difference between an LPA and LKA, other than in how many pipe rack frames were installed on platform decks in the cargo holds. The use of "combat loading' pretty much standardized the cargo layouts for Amphib ships. So, combining the classes started giving us LPH, LPD, LHD, and the like.
Now, I was Ambib (aka "gator navy") for only seven years (3 active 7& 4 Reserve); but, I've not heard of a US designation of LTH. But, there could be a Military Sealift designation along those lines; like one of the RoRo's.