We are renting a duplex and it is by a busy street we want to put up some fencing to keep 2 yr old grandson safe what could we erect that we could later take down and move with us? And how?:)

You will need to check codes in your town, your lease stipulations and with the landlord before you can put up any type of fence. Many home centers like Home Depot sell cyclone fence sections that are strung over a 4 sided framee. These are generally used for a dog run or a tempoary fence around construction sites. This is just an example and not meant to treat your grandchild like a dog. Dog run fencing

My landlord said it would be okay to erect a fence but my husband doesn't want to make it permanent as we would be paying for it. I have seen some sites where they use a short post that is pounded into ground to hold the post and another where they use a support for the post above ground I was wondering if they were strong enough to hold the sections upright?

The fence I showed you in the link needs no staking into the ground and gets its structural strength from its frame and how it connects at a 90 degree angle at each corner. It would be far more portable than anything needing stakes. Not sure how large you are thinking for the total area, a 2 year old can't go too far.