This is a bit easier to look at

How is that?

The problem is grade. All the added mulch in the world won't solve an underlying grade problem.

It appears that the mulched area where the puddles are evident is the local low area between your driveway and the adjacent lawn area. My guess from your description, and what appears to be a slope from the adjacent lawn area down towards your driveway, is the low area used to be adjacent to the driveway (the trench). It was probably put there to prevent exactly what you see happening now since you filled it.

To end the puddling you will need to remove the mulch and regrade the area to direct the water flow towards the street. Since your surrounding terrain is fairly flat, you may be limited on how much you can regrade, but in general you want at least 1" of fall per 10' of run.

So If I take up all the mulch an put more top soil down to level it all out will that fix it or do I need to make it slope towards the street? Once upon a time it all used to be level and mulched like we have it now, just whoever done it before dug it all up and created a trench. Before adding the mulch I had put top soil down to level it out, but it rained and the top soil still flooded over. How much do I need to put down to level it out etc.. I'm brand new to all this.. So your gonna have to dumb it down for me, also I would hate to dig up all the mulch I just put down..

To prevent standing water (puddles) you need some slope - water runs downhill. By leveling the area you may have taken away any slope that might have been there before.

To get some drainage going you need a drop of 1" in the underlying soil level for each 10 feet of run. First you figure out where you want the water to collect and where you want it to run to (to the street in this situation). You then grade the top of the soil surface so that water runs from all sides towards the collection zone and the collection zone is sloped downwards in the direction you want the water to flow and exit (towards the street).

The current edging block along the mulch edge may be acting as a bit of a dam, and it appears that the puddle area is the low spot from either side. It's not clear in the picture how the adjacent lawn area on the right side is sloped or how the sidewalk is sloped. If the sidewalk is sloping down from right to left as it heads towards the driveway, then the sidewalk will probably act as a dam. If that's the case, you probably won't be able to get flow out to the street without putting in a drain under the sidewalk.

If the terrain slopes steadily away from your driveway to the right in the picture, out past the end of the sidewalk end at the far right, and that is your property, then it would probably be better to regrade the entire area to direct the water run out to that far corner.

Just remember that for water to flow it needs a slope downhill. That slope needs to be 1" per 10'. Any low spots will fill and hold water until the underlying soil can absorb it, and that is your current problem - you have a low area with no place downhill for it to flow out. You will need to create that slope (grading) to achieve drainage. In a drainage zone, grass would be a better choice than mulch - it will stabilize the soil, filter the runoff somewhat, and it won't wash away.