begin

She began singing.
She began to sing.

When "begin" is used in non-continuous tenses, you can either use a gerund or an infinitive.

She is beginning to sing.
 

When "begin" is used in continuous tenses, an infinitive is used.

forget

She forgot reading the book when she was a kid.

When "forget" is used with a gerund, it means "to forget that you have done something." The sentence above means that she read  the book when she was a kid, and that she has forgotten that fact.

She forgot to pay the rent this month.
 

When forget is used with an infinitive, it means "to forget that you need to do something." The sentence above means that she forgot that she needed to pay the rent.

keep

She kept talking.

"Keep" is normally used with a gerund to mean that you continue doing an action.

The attackers kept hostages to prevent the police from entering.
 

"Keep" can also be used with an object followed by an  infinitive, but then the infinitive takes on the meaning of "in order  to... ." In the sentence above, the attackers kept hostages in order to prevent the police from entering.

mean

If you take that job in London, it will mean travelling for two hours every day.

Expresses what the result of an action will be. It is a synonym of “involve” or “require”.

I meant to phone your mom, but my cell didn't work.

Sorry, I didn't mean to hurt you.

Expresses an intention/plan. It is a synomym of “expect” or “want”

need

The house needs cleaning.

When "need" is used with a gerund, it takes on a passive meaning. The sentence above means "the house needs to be cleaned."

He needs to call his boss.
He needs him to call his boss.

"Need" is usually used with an infinitive or an object + an infinitive.

regret

I regretted being late to the interview.
 

"Regret" is normally used with a gerund.

We regret to inform you that your position at the company is being eliminated.

"Regret" is sometimes used with infinitives such as "to inform." In the sentence above, "We regret to inform you" means "We wish we did not have to tell you (bad news)."

remember

I remember mentioning the meeting yesterday.

When "remember" is used with a gerund, it means "to remember that you have done something." The sentence above means that I mentioned the meeting, and that I remember the fact that I did that.

He remembered to turn off the lights before he left.
 

When "remember" is used with an infinitive, it means "to remember that you need to do something." The sentence above means that he remembered that he needed to turn the lights off.

start

Marge started talking really fast.
Marge started to talk really fast.

When "start" is used in non-continuous tenses, you can either use a gerund or an infinitive.

Marge is starting to talk really fast.

When "start" is used in continuous tenses, an infinitive is used.

I started to learn Russian, but it was so much work that I finally quit the class.

In other situations, an infinitive means that you did not complete or continue an action. 

stop

He stopped smoking for health reasons.

"Stop" is normally used with a gerund.

He stopped to rest for a few minutes.

When "stop" is used with an infinitive, the infinitive takes on the meaning of "in order to." In the sentence above, he stopped in order to rest for a few minutes.

try
 

She can't find a job. She tried looking in the paper, but there was nothing. She tried asking friends and family, but nobody knew of anything. She also tried going shop to shop, but nobody was hiring.

"Try + gerund" means to try or to experiment with different methods to see if something works.

She tried eating the snake soup, but she didn't like it.

"Try + gerund" is often used when you experiment with something, but you do not really like it or want to do it again.

She tried to climb the tree, but she couldn't even get off the ground.
 

When you "try to do" something, you want to do it, but you do
  not succeed in actually doing it. In the sentence above, an infinitive is
  used because she cannot successfully climb the tree.

Try not to wake the baby when you get up tomorrow at 5 AM.

An infinitive is also used if you are asking someone to try something they may or may not be able to accomplish.