Dear student,
Partner by estoppel is one who by his conduct allow him to be represented as a partner of a firm. If anyone advances money to the firm on the basis of his representation, then he becomes liable to that third party and cannot deny its claim.
Partner by holding out is the one who does not deny when he is declared as a partner of a firm. If anyone advances money to the firm on the basis of his declaration, then he becomes liable to that third party and cannot deny its claim.
Both the partner by estoppel and partner by holding out are nominal partners. They just differ on a point that the partner by estoppel represent himself as a partner while partner by holding out is declared as a partner.
Regards