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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 odeel / odela.
I want him to leave. - Chci, aby odeel.
I want the weather to be nice. - Chci, aby bylo hezky (aby bylo hezk poas).
I want us to be friends. - Chci, abychom byli ptel.
I want you (plural) to be happy. - Chci, abyste byli astni.
I want them to stop talking. - Chci, aby pestali mluvit.

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

singular	plural
1. (j) 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 odeel.")

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 chtt (1st person, future tense), abys to vdl.

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

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

Other similar constructions with "aby":

I told him to do that. - ekl jsem mu, aby to udlal.
He asked me to help him. - Podal m, abych mu pomohl.

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