The formation of unstaggered localized modes in dynamical lattices can be supported by the interplay of discreteness and nonlinearity with a finite relaxation time. In rapidly responding nonlinear media, on-site discrete solitons are stable, and their broad inter-site counterparts are marginally stable, featuring a virtually vanishing real instability eigenvalue. The solitons become unstable in the case of the slowly relaxing nonlinearity. The character of the instability alters with the increase of the delay time, which leads to a change in the dynamics of unstable discrete solitons. They form robust localized breathers in rapidly relaxing media, and decay into oscillatory diffractive pattern in the lattices with a slow nonlinear response. Marginally stable solitons can freely move across the lattice. The delay of the medium's response is unavoidable in natural phenomena and technology. In this respect, it is relevant to mention the cellular physiology, genetics, signal processing, transport of impulses through neural networks, the propagation of light signals in optical resonator systems and cavities, and so on. The delayed response may significantly affect many phenomena in these fields. In particular, essential issues, which are addressed in this paper, are the existence, stability and dynamics of localized modes in dynamical lattices with the non-instantaneous nonlinear response. The lattice, which may describe an array of nonlinear optical waveguides, is modeled by the timedelayed discrete nonlinear Schrödinger equation. The localized modes in this model are explored using a properly modified delay-differential equation solver package DDE-BIFTOOL v. 2.00. We show that the discrete bright on-site and inter-site solitons (so classified according to the location of their center) can be created by the modulational instability of continuous waves, which occurs in the same parameter region as in the lattice with the instantaneous nonlinearity. However, the stability of the solitons is affected by the delayed response in three different ways: (i) the solitons are destroyed when the delay time exceeds a certain critical value; (ii) in the case of the fast nonlinear response (short temporal delay), the instability growth rate of inter-site solitons decreases in comparison with the instantaneous model; (iii) the instability type changes for intermediate values of the delay time. The inter-site and on-site solitons with very close values of the power in the fast-responding media evolve into localized breathing modes when their amplitude is slightly perturbed. They can also be transformed into moving localized modes by the application of a kick. However, larger values of the delay time enhance the temporal correlation between time-distant events, which results in bringing moving solitons to a halt.