I have downloaded two Landsat 5 images because my study area is divided between the 2 tiles. I have created a composite of all the bands and then tried the mosaic tool in ArcMap but the result I have contains the black border that the raw images have and it is not correct. Is there a way how I could connect the 2 images in order to have only one image in the end to continue with my processing?
This 'black border' is probably 'background value' (normally RGB: 0,0,0). You can check the value of this 'black border' by using the Identify tool and clicking on it.
If I understood correctly, you need to remove or mask this background first before doing your mosaic layer by for example 'Mosaic to New Raster'.
One option, (if it is just about not seeing the background(: right-click on the composite layer and go to Layer properties/Symbology/RGB Composite and check: Display Backround Value as transparent.
Another option, (if you want to ignore this background pxels while processing and avoid processing errors of overlaping areas) is setting the target values as 'NoData'. You can easily achieve this by opening the ArcGIS Catalog as explained here. Right-click on your composite file (multiband raster) and go to General/ Raster Information. Edit the NoData Value and set it as desired (probably RGB: 0,0,0). An alternative method would be exporting to a new raster layer and setting the 'NoData' value to 0. From now on, ArcGIS tools should ignore this pixels.