Adapters are oligos that are attached to the ends of nucleic acid fragments to create "libraries". These libraries (inserts) is what gets sequenced. Barcodes on the other hand are much shorter stretches of nucleotides that are used to label individual samples so multiple samples can be pooled together for sequencing. Illumina adapters (from Illumina and other vendors) include "barcodes' in the adapter itself (there is a core sequence followed by barcodes.

You can find a PDF from Illumina that contains the sequences here.

May I use another tool other than trimmomatic?

Sure. I recommend bbduk.sh from BBMap suite. A guide is available here. There is also cutadapt, sickle and others.