First, keep in mind that these descriptions are intentionally kept vague and generic in the rules, to keep them compatible with a variety of interpretations and/or settings. Official settings sometimes go into more detail (e.g. Forgotten Realms has the weave), but also sometimes don’t (e.g. Eberron doesn’t really detail the mystical mechanics of magic much beyond the default).

Second, for the most part, these distinctions are not made very important to the games—it is enough to know that what a cleric does is different from what a wizard does. There presumably are some pretty significant technical differences for actual practitioners, but players don’t really need to know them (and thus designers don’t need to invent them). Different classes of ability (e.g. Extraordinary, Supernatural, Spell-like, Spell, etc.) cover the actual rules in generic ways that apply to many different effects. Individual descriptions in classes are sufficient to cover any other distinctions.

With those in mind, here are the concepts of various forms of magic.

Arcane—playing with the cosmic API

Arcane spellcasters cast spells by interacting directly with reality. Whether by quirk of chance or conscious divine design, there are certain combinations of words, objects, chants, and gestures that produce various effects known as spells.

Arcane spellcasters learn these combinations, and can then perform them to produce the corresponding effects. Higher level spells are those with more complex patterns, harder to hold onto. And holding onto them is hard: an incomplete pattern, ready to be finished, that is, cast, is actually a part of the person who has started it, and he or she is limited in how many of these he or she can handle at once. These incomplete patterns can be detected and effected (e.g. by the spellthief class).

This style of magic is based heavily in the writings of Jack Vance, if you seek more detailed illustrations of how this works.

Now then, particular classes interact with these patterns in different ways. Wizards laboriously study them, and each morning can prepare a certain number, leaving them ready to be cast. This is the most directly Vancian in 3.5.

But sorcerers instead have these patterns as indelible parts of themselves, innately accessing these patterns and able to complete them without necessarily knowing them consciously. Often, the source of this innate connection to the pattern is due to magical ancestry, particularly draconic, since dragons are powerful natural sorcerers, and Draconic is often used as the language of magic.

And bards tap into the patterns through music, empowering them not with knowledge or blood, but heart and soul. They know music, but the magic comes from following their inspiration.

There are other arcane spellcasters. Assassins cast as sorcerers do, but using Intelligence. Presumably they simply ingrain knowledge of a select few spells as indelibly on themselves as sorcerers’ blood does on them. There is even one class, the dark hunter, that cats arcane spells based on Wisdom, able to sense, perceive, and understand these cosmic patterns.

And then there are warlocks: arcane, but no spells. Their connection to these patterns are much tighter, often forged by a deal with fey or fiend, and these deals often affect entire lineages. Warlocks can cast invocations at will, with no preparation or limited spell slots.

Shadow—the shadow of the arcane

Shadow magic is basically arcane, but manipulates the “shadows” of arcane patterns. This makes shadow magic somewhat more meta, more able to manipulate magic itself, but also makes it more difficult to use and less efficient.

Divine—spells as gifts

Divine spellcasters do not master their own spells: their own patterns, which they need only complete to cast the spell, are granted to them in response to prayer.

Divine spellcasters pray to different things, but the gifting of spells is the same whether you pay to a god, or nature itself, or even just a great cosmic ideal like good or evil or elves (seriously).

Also, paladins are lawful, but their devotion is first and foremost to Good. Paladins must be lawful because they are to be a shining example of goodness, and that means being always honorable and above-board, but their purpose is always good.

Infusions—item-magic

Infusions are used by the artificer class from Eberron Campaign Setting. They are just spells that can only affect objects or constructs, not flesh or soul. The spells are neither arcane nor divine, but involve similar patterns to both. Like the bard is inspired to magic by the forms of music, the practice of construction gives an artificer insight into the patterns of magic.

Psionics—power of the self

Where arcane magic utilizes the patterns inherent in the magical universe to produce spell effects, and divine spellcasters have spells just given to them, psionic manifesters are just straight-up enforcing their will on reality. “I reject your reality and substitute my own,” is very literally their motto.

Different manifesters enforce their will in different ways, of course; that’s why there are different classes and different ability scores used. But they all revolve around this premise.

Pacts—sharing your soul for fun and profit

Since you suggested pacts were like divine magic, no. Gods are empowered by belief and prayer, and grant spells to their faithful so they can perform miracles in that god’s name, spreading their faith and garnering them more believers. The vestiges described in Tome of Magic are not gods—in fact, they don't even really exist, at least in this reality. They are the “vestiges” of powers from the past, some gods, some fiends, some just powerful mortals, and some just unlucky. They linger in some not-place, beyond the reach of reality—gods included. Which makes the gods fear and hate them.

And they are desperate for the opportunity, however fleeting, to experience reality again. So binders allow these vestiges into their own souls, to let them share their lives. The vestiges always try for more influence, of course, but ultimately the binder is the party in control here. And the vestiges have to grant powers to attract binders in the first place.

Mechanically, vestige-granted abilities are Supernatural, which has various implications for interacting with things. They are not spells, and are generally continuous or at-will, though a few have five-round cooldowns.

Truenaming—Continuing creation

Truenaming works by making statements of fact in the language of creation: anything you say in this language becomes true.

Unfortunately, this chapter of Tome of Magic was awful, and the system is nonfunctional. Several excellent homebrew replacements exist, though.

Incarnum—Soul-stuff sculpting

Incarnum is the power of soul-stuff. Not souls, but the stuff souls are made from. The stuff that was once a soul or will one day be a soul, but is currently just... stuff. Incarnum. It’s blue.

Anyway, incarnum meldshapers shape and invest this soulstuff into what amount to temporary magical items. Incarnates get useful tools, totemists get the forms of rending claws or fearsome jaws or whatever. Soulborns get... little and less, and should just be ignored.

There’s also necrocarnum, which does use actual souls, specifically souls tortured and flayed into shapeable soulstuff. As you might guess, necrocarnum use is one of the most blatantly evil things in the system.

Sublime—blade “magic”

The maneuvers in Tome of Battle are generally non-magical, though a few are Supernatural. These are called “magic” only because they can seem like magic to those less dedicated to the sublime martial arts. Even those that are supernatural are much more like the monk’s supernatural abilities, just “mystical martial arts.”