Why is it 'no veo a nadie' and not 'veo a nadie'?

Because Spanish uses a double negative. When a negative word like nadie, nada or ninguno comes after the verb, you must also put no before the verb: no veo a nadie (I don't see anyone). The two negatives do not cancel out, as they would in formal English; together they make a single negative statement. The only way to drop the no is to move the negative word in front of the verb: a nadie veo, nadie viene. Before the verb, one negative is enough; after it, you need both.

What is the difference between algún and alguno?

They are the same word; algún is the shortened form used directly before a masculine singular noun. Algún día (some day), but the full alguno comes back when it stands alone or follows the noun: ¿tienes alguno? (do you have any?). The feminine alguna never shortens: alguna idea. The same apocopation hits ninguno, which becomes ningún before a masculine singular noun: ningún problema.

Does cualquiera mean 'anyone' or 'any'?

Both, depending on position. Before a noun it shortens to cualquier and means 'any (at all)': cualquier día (any day), cualquier persona (any person). Standing alone it is cualquiera and means 'anyone' or, with a faint sneer, 'any old one': eso lo hace cualquiera (anyone can do that). Watch the tone - calling something un trabajo cualquiera means a run-of-the-mill, nothing-special job.

Spanish Indefinites

Indefinites point at quantity or identity without pinning it down: some, none, someone, no one, something, nothing, any. Spanish keeps a small, regular set, and most of the difficulty is in two patterns English does not share: agreement plus apocopation, and the double negative.

The cleanest way in is to sort them by what they refer to.

The set, sorted

Refers to	"some / a"	"none / no"	English
A person (alone)	alguien	nadie	someone / no one
A thing (alone)	algo	nada	something / nothing
A countable noun	alguno (algún)	ninguno (ningún)	some / none, any
Whatever / any	cualquiera (cualquier)	-	any (at all)
The whole lot	todo	-	all, every
A different one	otro	-	another, other

Alguien, nadie, algo and nada are invariable. They never change for gender or number. Alguno, ninguno, cualquiera, todo and otro do agree, and that is where the work is.

Agreement and the disappearing vowel

Alguno and ninguno behave like ordinary adjectives - except for one quirk. Directly before a masculine singular noun they drop the final -o and take a written accent:

algún día (some day) - not alguno día
ningún problema (no problem) - not ninguno problema

The accent matters; it is not optional. The full forms return the moment the word stands alone or follows its noun:

No tengo ninguno. (I don't have any.)
¿Conoces alguno? (Do you know any?)

The feminine and plural forms never shorten:

alguna idea (some idea), algunas personas (some people)
ninguna duda (no doubt at all)

A note on ninguno: in the singular it is overwhelmingly more common than the plural. Spanish says no tengo ningún amigo aquí (I have no friends here) with the singular, where English reaches for a plural.

Todo agrees fully and often pairs with the article: todo el día (all day), toda la noche (all night), todos los días (every day). Otro agrees too, and famously takes no article in front: otro café (another coffee), never un otro café.

The double negative

This is the rule English speakers fight hardest, because school English bans it. Spanish requires it.

When nadie, nada, ninguno (or nunca, tampoco) come after the verb, the verb must still be negated with no:

No veo a nadie. (I don't see anyone. Literally: I don't see no one.)
No tengo nada. (I have nothing.)
No hay ningún problema. (There is no problem at all.)

Two negatives, one negative meaning. They reinforce; they do not cancel.

The one way to drop the no is to move the negative word in front of the verb. Then one negative is enough:

Nadie viene. (No one is coming.)
Nada me gusta. (Nothing pleases me.)
A nadie le importa. (Nobody cares.)

So the rule is positional. Negative-before-verb: one negative. Negative-after-verb: no plus the negative. Both are correct; no veo a nadie and a nadie veo mean the same thing.

Note the personal a in no veo a nadie and busco a alguien: when the indefinite stands for a person and is a direct object, the personal a still applies.

Cualquiera: any old one

Cualquiera is the trickiest of the set because its meaning shifts with its tone and position.

Before a noun it shortens to cualquier (for both genders) and means any at all, with no shortlisting:

Cualquier día es bueno. (Any day is fine.)
Llama a cualquier hora. (Call at any time.)

Standing alone, the full cualquiera means anyone or whichever one:

Eso lo hace cualquiera. (Anyone can do that.)
Toma uno cualquiera. (Take any one of them.)

Here is the trap. Un ... cualquiera after a noun carries a dismissive shrug - any old one, nothing special:

un trabajo cualquiera (a run-of-the-mill job)
una persona cualquiera (a nobody, just anyone)

So cualquier trabajo (any job, your choice) and un trabajo cualquiera (a nothing job) are not the same. Position changes the colour of the word entirely.

Worked examples

¿Hay alguien en casa? - No, no hay nadie. (Is anyone home? No, there's no one.)
¿Quieres algo? - No, no quiero nada. (Do you want something? No, I don't want anything.)
Tengo algunos amigos aquí, pero ningún familiar. (I have some friends here, but no relatives.)
Cualquiera puede aprender español; cualquier persona, a cualquier edad. (Anyone can learn Spanish; any person, at any age.)

Common mistakes English speakers make

Cancelling the double negative. Hearing no veo a nadie as "I don't see nobody" and "correcting" it to veo a nadie is wrong. Spanish needs both halves.

Forgetting to shorten. Alguno día and ninguno problema sound off. Before a masculine singular noun it is algún día, ningún problema.

Pluralising ninguno. English "I have no friends" tempts a plural, but Spanish prefers the singular: no tengo ningún amigo.

Slipping un in front of otro. "Another one" is otro, never un otro.