You can check in to a us website by USGS. there are bunch of Lansat images from 1970s to date i think at 30 m resolution and last time i checked it was free and it covers almost the whole world. just follow the instructionsthe adress i think is usgs.gov
You will get Landsat images at glovis.usgs.gov. satellite images from 1972 to present... This si best as a starting point. If you want Indian Satellite images some of them are available in bhuvan website.
It depends which satellite sensors you need data from, and whether you want processed products (e.g. Surface Temperature), or raw satellite data (counts or derived reflectances/brightness temperatures). Most repositories require you to register for an account, but it's usually free if you need the data for research purposes. I use data from EUMETSAT (http://www.eumetsat.int/website/home/Data/Products/index.html), from NERC's repository (http://data-search.nerc.ac.uk/search/full), from BADC (http://badc.nerc.ac.uk/home/index.html) and from MODIS's own pages (http://modis.gsfc.nasa.gov/data/).
If you are looking into marine/oceanographic remote sensing data, NOAA's Ocean Watch is one of the first I always go to http://oceanwatch.pifsc.noaa.gov/las/servlets/dataset?catitem=0
Thank you very much to Rishita Rangarh, Nura Kura, Ion Andronache, Niladri Gupta, Jobin Thomas, Jose Gomez Castaño, Shona Mackie, Nathan Furey for Responding my Question
The satellite images that are free and I have used are Landsat and MODIS. Landsat with a spatial resolution of 30 m and a temporal resolution of 15-16 days. MODIS, I use the products they offer, depending on the product is its spatial and temporal resolution, the bands may be available online every day, according to the literature (I have not used yet). Here is the link MODIS products: https://lpdaac.usgs.gov/products/modis_data_products_table
All the mentioned sources are indeed good ones for archived satellite data. However, one may look for recent launches of satellites. Mostly, with the recent launches, the launching agencies offer few satellite images free of cost through 'windows of opportunity', through which the user may procure the images of his/her choice.