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The conjunction "aby" means "to, in order to, so that". Besides its other use, it is also used to form constructions like "I want something to happen" or "I want someone to do something". Examples:

I want you to leave. - Chci, abys odešel / odešla.
I want him to leave. - Chci, aby odešel.
I want the weather to be nice. - Chci, aby bylo hezky (aby bylo hezké počasí).
I want us to be friends. - Chci, abychom byli přátelé.
I want you (plural) to be happy. - Chci, abyste byli šťastni.
I want them to stop talking. - Chci, aby přestali mluvit.

The word "aby" changes based on the person it refers to:

singular	plural
1. (já) abych	1. (my) abychom
2. (ty) abys	2. (vy) abyste
3. (on/ona/ono) aby	3. (oni/ony/ona) aby

The verb that is used in the sentence that begins with "aby" is always in the conditional form, which looks exactly like the past tense (e.g. "Chci, abys odešel.")

The "aby" constructions always follow this pattern, no matter in what person or tense the verb in the first sentence is. Examples:

I will want you to know that.
Budu chtít (1st person, future tense), abys to věděl.

They wanted you to know that.
Chtěli (3rd person plural, past tense), abys to věděl.

We would like you to know that.
Chtěli bychom (1st person plural, conditional), abys to věděl.

Other similar constructions with "aby":

I told him to do that. - Řekl jsem mu, aby to udělal.
He asked me to help him. - Požádal mě, abych mu pomohl.

Note: Remember to always place a comma in front of "aby".